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Accuracy and Stability of Numerical Algorithms
Nicholas J. Higham University of Manchester Manchester, England
Accuracy and Stability of Numerical Algorithms
Society for Industrial and Applied Mathematics Philadelphia
Copyright © 1996 by the Society for Industrial and Applied Mathematics. 1098765432 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information. write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688.
Library of Congress Cataloging-in-Publication Data Higham, Nicholas J., 1961Accuracy and stability of numerical algorithms / Nicholas J. Higham. p. cm. Includes bibliographical references (p. - ) and index. ISBN O-8987 l-355-2 (pbk.) 1. Numerical analysis--Data processing. 2. Computer algorithms. I. Title. QA297.H53 1996 5 19.4’0285’5 1 --dc20 95-39903
o is a registered trademark.
Dedicated to Alan M. Turing and James H. Wilkinson
Contents
List of Figures
xvii
List of Tables
xix
Preface
xxi xxvii
About the Dedication
1 Principles of Finite Precision Computation 1.1 Notation and Background . . . . . . . . . . . . . . . . . . . . 1.2 Relative Error and Significant Digits . . . . . . . . . . . . . . 1.3 Sources of Errors . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Precision Versus Accuracy . . . . . . . . . . . . . . . . . . . . 1.5 Backward and Forward Errors . . . . . . . . . . . . . . . . . . 1.6 Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Solving a Quadratic Equation . . . . . . . . . . . . . . . . . . 1.9 Computing the Sample Variance . . . . . . . . . . . . . . . . . 1.10 Solving Linear Equations . . . . . . . . . . . . . . . . . . . . . 1.10.1 GEPP Versus Cramer’s Rule . . . . . . . . . . . . . . . 1.11 Accumulation of Rounding Errors . . . . . . . . . . . . . . . . 1.12 Instability Without Cancellation . . . . . . . . . . . . . . . . . 1.12.1 The Need for Pivoting . . . . . . . . . . . . . . . . . . 1.12.2 An Innocuous Calculation? . . . . . . . . . . . . . . . . 1.12.3 An Infinite Sum . . . . . . . . . . . . . . . . . . . . . . 1.13 Increasing the Precision . . . . . . . . . . . . . . . . . . . . . . 1.14 Cancellation of Rounding Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14.1 Computing 1.14.2 QR Factorization . . . . . . . . . . . . . . . . . . . . . 1.15 Rounding Errors Can Be Beneficial . . . . . . . . . . . . . . . 1.16 Stability of an Algorithm Depends on the Problem . . . . . . 1.17 Rounding Errors Are Not Random . . . . . . . . . . . . . . . 1.18 Designing St able Algorithms . . . . . . . . . . . . . . . . . . . 1.19 Misconceptions . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 2 4 5 7 7 9 10 11 12 13 14 16 17 17 17 18 19 21 22 24 26 27 29 30 31
V111 1.20 Rounding Errors in Numerical Analysis . . . . . . . . . . . . . 1.21 Notes and References . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32 32 36
39 2 Floating Point Arithmetic Floating Point Number System . . . . . . . . . . . . . . . . . 40 2.1 Model of Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . 44 2.2 IEEE Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3 2.4 Aberrant Arithmetics . . . . . . . . . . . . . . . . . . . . . . . 48 Choice of Base and Distribution of Numbers . . . . . . . . . . 51 2.5 Statistical Distribution of Rounding Errors . . . . . . . . . . . 52 2.6 53 Alternative Number Systems . . . . . . . . . . . . . . . . . . . 2.7 Accuracy Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.8 Notes and References . . . . . . . . . . . . . . . . . . . . . . . 56 2.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3
Basics Inner and Outer Products . . . 3.1 The Purpose of Rounding Error 3.2 Running Error Analysis . . . . . 3.3 Notation for Error Analysis . . 3.4 Matrix Multiplication . . . . . . 3.5 Complex Arithmetic . . . . . . . 3.6 Miscellany . . . . . . . . . . . . 3.7 Error Analysis Demystified . . . 3.8 Other Approaches . . . . . . . . 3.9 3.10 Notes and References . . . . . . Problems . . . . . . . . . . . . .
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4 Summation 4.1 Summation Methods . . . . . . . Error Analysis . . . . . . . . . . . 4.2 Compensated Summation . . . . . 4.3 Other Summation Methods . . . . 4.4 Statistical Estimates of Accuracy 4.5 Choice of Method . . . . . . . . . 4.6 Notes and References . . . . . . . 4.7 Problems . . . . . . . . . . . . . .
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67 . 68 . 71 . 72 . 73 . 76 . 78 . 80 . 82 . 83 . 84 . 84
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87 . . . . 88 . . . . 89 . . . . 92 . . . . 97 . . . . 98 . . . . 98 . . . . 100 . . . . 100
103 5 Polynomials Horner‘s Method . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.1 Evaluating Derivatives . . . . . . . . . . . . . . . . . . . . . . 106 5.2 The Newton Form and Polynomial Interpolation . . . , . . . . 109 5.3
ix 5.4
Notes and References . . . . . . . . . . . . . . . . . . . . . . . 113 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
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117 118 120 124 126 127
7 Perturbation Theory for Linear Systems Normwise Analysis . . . . . . . . . . . . . . . . . . . . . . . . 7.1 7.2 Componentwise Analysis . . . . . . . . . . . . . . . . . . . . . 7.3 Scaling to Minimize the Condition Number . . . . . . . . . . . 7.4 The Matrix Inverse . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Numerical Stability . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Practical Error Bounds . . . . . . . . . . . . . . . . . . . . . . 7.8 Perturbation Theory by Calculus . . . . . . . . . . . . . . . . Notes and References . . . . . . . . . . . . . . . . . . . . . . . 7.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
131 132 134 137 140 140 141 142 144 145 147
8 Triangular Systems 8.1 Backward Error Analysis . . 8.2 Forward Error Analysis . . . Bounds for the Inverse . . . 8.3 8.4 A Parallel Fan-In Algorithm 8.5 Notes and References . . . . 8.5.1 LAPACK . . . . . . . Problems . . . . . . . . . . .
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151 152 155 159 162 164 166 166
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169 170 174 177 181 183 186 191 192 194 195 198
6 Norms Vector Norms . . . . 6.1 6.2 Matrix Norms . . . . 6.3 The Matrix p-Norm . 6.4 Notes and References Problems . . . . . . .
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9 LU Factorization and Linear Equations Gaussian Elimination . . . . . . . . . . 9.1 Error Analysis . . . . . . . . . . . . . . 9.2 9.3 The Growth Factor . . . . . . . . . . . Special Matrices . . . . . . . . . . . . . 9.4 9.5 Tridiagonal Matrices . . . . . . . . . . Historical Perspective . . . . . . . . . . 9.6 Scaling . . . . . . . . . . . . . . . . . . 9.7 9.8 A Posteriori Stability Tests . . . . . . . Sensitivity of the LU Factorization . . 9.9 9.10 Notes and References . . . . . . . . . . 9.10.1 LAPACK . . . . . . . . . . . . .
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X Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 10 Cholesky Factorization 10.1 Symmetric Positive Definite Matrices . . . . 10.1.1 Error Analysis . . . . . . . . . . . . . 10.2 Sensitivity of the Cholesky Factorization . . 10.3 Positive Semidefinite Matrices . . . . . . . . 10.3.1 Perturbation Theory . . . . . . . . . 10.3.2 Error Analysis . . . . . . . . . . . . . 10.4 Symmetric Indefinite Matrices and Diagonal Method. . . . . . . . . . . . . . . . . . . . . 10.4.1 Complete Pivoting . . . . . . . . . . 10.4.2 Partial Pivoting . . . . . . . . . . . . 10.5 Nonsymmetric Positive Definite Matrices . . 10.6 Notes and References . . . . . . . . . . . . . 10.6.1 LAPACK . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . .
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203 204 205 209 210 211 214
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218 219 221 223 224 226 227
11 Iterative Refinement 11.1 Convergence of Iterative Refinement . 11.2 Iterative Refinement Implies Stability 11.3 Notes and References . . . . . . . . . 11.3.1 LAPACK. . . . . . . . . . . . Problems . . . . . . . . . . . . . . . .
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231 232 235 241 243 243
12 Block LU Factorization 12.1 Block Versus Partitioned LU Factorization . . . . . . . . . . . 12.2 Error Analysis of Partitioned LU Factorization . . . . . . . . . 12.3 Error Analysis of Block LU Factorization . . . . . . . . . . . . 12.3.1 Block Diagonal Dominance . . . . . . . . . . . . . . . . 12.3.2 Symmetric Positive Definite Matrices . . . . . . . . . . 12.4 Notes and References . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 LAPACK . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
245 246 248 250 251 255 257 258 258
13 Matrix Inversion 13.1 Use and Abuse of the Matrix Inverse . . . . 13.2 Inverting a Triangular Matrix . . . . . . . . 13.2.1 Unblocked Methods . . . . . . . . . . 13.2.2 Block Methods . . . . . . . . . . . . 13.3 Inverting a Full Matrix by LU Factorization 13.3.1 Method A . . . . . . . . . . . . . . . 13.3.2 Method B . . . . . . . . . . . . . . .
261 262 265 265 267 270 270 271
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xi 13.3.3 Method C . . . . . 13.3.4 Method D . . . . . 13.3.5 Summary. . . . . . 13.4 Gauss-Jordan Elimination 13.5 The Determinant . . . . . 13.5.1 Hyman’s Method . 13.6 Notes and References . . . 13.6.1 LAPACK . . . . . . Problems . . . . . . . . . .
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272 273 275 275 281 282 283 285 285 289
14 Condition Number Estimation 14.1 H o w t o E s t i m a t e C o m p o n e n t w i s e C o n d i t i o n Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 The p-Norm Power Method. . . . . . . . . . . . . . . 14.3 LAPACK l-Norm Estimator . . . . . . . . . . . . . . 14.4 Other Condition Estimators . . . . . . . . . . . . . . 14.5 Condition Numbers of Tridiagonal Matrices . . . . . 14.6 Notes and References . . . . . . . . . . . . . . . . . . 14.6.1 LAPACK . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . .
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290 291 294 297 301 304 306 306
15 The Sylvester Equation 15.1 Solving the Sylvester Equation . 15.2 Backward Error . . . . . . . . . 15.2.1 The Lyapunov Equation 15.3 Perturbation Result . . . . . . . 15.4 Practical Error Bounds . . . . . 15.5 Extensions . . . . . . . . . . . . 15.6 Notes and References . . . . . . 15.6.1 LAPACK . . . . . . . . . Problems . . . . . . . . . . . . .
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309 311 313 316 318 320 321 322 324 324
16 Stationary Iterative Methods 16.1 Survey of Error Analysis . . . . . 16.2 Forward Error Analysis . . . . . . 16.2.1 Jacobi’s Method . . . . . . 16.2.2 Successive Overrelaxation 16.3 Backward Error Analysis . . . . . 16.4 Singular Systems . . . . . . . . . 16.4.1 Theoretical Background . 16.4.2 Forward Error Analysis . . 16.5 Stopping an Iterative Method . . 16.6 Notes and References . . . . . . .
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325 327 329 332 334 334 336 336 338 341 343
xii Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
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345 346 353 358 358 359
18 QR 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8 18.9
Factorization Householder Transformations . . . . . . . . . QR Factorization . . . . . . . . . . . . . . . Error Analysis of Householder Computations Aggregated Householder Transformations . . Givens Rotations . . . . . . . . . . . . . . . Iterative Refinement . . . . . . . . . . . . . . Gram-Schmidt Orthogonalization . . . . . . Sensitivity of the QR Factorization . . . . . Notes and References . . . . . . . . . . . . . 18.9.1 LAPACK . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . .
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361 362 363 364 370 371 375 376 381 383 386 387
19 The 19.1 19.2 19.3 19.4 19.5 19.6 19.7 19.8 19.9
Least Squares Problem Perturbation Theory . . . . . . . . . . . . . . . . Solution by QR Factorization . . . . . . . . . . . Solution by the Modified Gram-Schmidt Method The Normal Equations . . . . . . . . . . . . . . . Iterative Refinement . . . . . . . . . . . . . . . . . The Seminormal Equations . . . . . . . . . . . . . Backward Error . . . . . . . . . . . . . . . . . . . Proof of Wedin’s Theorem . . . . . . . . . . . . . Notes and References . . . . . . . . . . . . . . . . 19.9.1 LAPACK . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . .
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391 392 395 396 397 399 403 404 407 409 412 412
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415 416 417 419 422 423 423
17 Matrix Powers 17.1 Matrix Powers in Exact Arithmetic . . 17.2 Bounds for Finite Precision Arithmetic 17.3 Application to Stationary Iteration . . 17.4 Notes and References . . . . . . . . . . Problems . . . . . . . . . . . . . . . . .
20 Underdetermined Systems 20.1 Solution Methods . . . . 20.2 Perturbation Theory . . 20.3 Error Analysis . . . . . . 20.4 Notes and References . . 20.4.1 LAPACK . . . . . Problems . . . . . . . . .
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X111 21 Vandermonde Systems 21.1 Matrix Inversion . . . . . . . . . 21.2 Primal and Dual Systems . . . . 21.3 Stability . . . . . . . . . . . . . 21.3.1 Forward Error . . . . . . 21.3.2 Residual . . . . . . . . . 21.3.3 Dealing with Instability . 21.4 Notes and References . . . . . . Problems . . . . . . . . . . . . .
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425 426 428 434 435 437 438 440 441
22 Fast Matrix Multiplication 22.1 Methods . . . . . . . . . . . . . 22.2 Error Analysis . . . . . . . . . . 22.2.1 Winograd’s Method . . . 22.2.2 Strassen’s Method . . . . 22.2.3 Bilinear Noncommutative 22.2.4 The 3M Method . . . . . 22.3 Notes and References . . . . . . Problems . . . . . . . . . . . . .
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445 446 450 451 452 456 458 459 461
Fast Fourier Transform and Applications The Fast Fourier Transform . . . . . . . . . . . . . . . . . . . Circulant Linear Systems . . . . . . . . . . . . . . . . . . . . . Notes and References . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
465 466 468 470 471
23 The 23.1 23.2 23.3
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24 Automatic Error Analysis 24.1 Exploiting Direct Search Optimization 24.2 Direct Search Methods . . . . . . . . . 24.3 Examples of Direct Search . . . . . . . 24.3.1 Condition Estimation . . . . . . 24.3.2 Fast Matrix Inversion . . . . . . 24.3.3 Solving a Cubic . . . . . . . . . 24.4 Interval Analysis . . . . . . . . . . . . . 24.5 Other Work . . . . . . . . . . . . . . . 24.6 Notes and References . . . . . . . . . . Problems . . . . . . . . . . . . . . . . .
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25 Software Issues in Floating Point Arithmetic 25.1 Exploiting IEEE Arithmetic . . . . . . . . . . 25.2 Subtleties of Floating Point Arithmetic . . . . 25.3 Cray Peculiarities . . . . . . . . . . . . . . . . 25.4 Compilers . . . . . . . . . . . . . . . . . . . .
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473 474 477 479 480 481 483 485 487 489 490 491 492 495 496 497
xiv Determining Properties of Floating Point Arithmetic . . . . . Testing a Floating Point Arithmetic . . . . . . . . . . . . . . . Portability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.7.1 Arithmetic Parameters . . . . . . . . . . . . . . . . . . 25.7.2 2×2 Problems in LAPACK . . . . . . . . . . . . . . . 25.7.3 Numerical Constants . . . . . . . . . . . . . . . . . . . 25.7.4 Models of Floating Point Arithmetic . . . . . . . . . . 25.8 Avoiding Underflow and Overflow . . . . . . . . . . . . . . . . 25.9 Multiple Precision Arithmetic . . . . . . . . . . . . . . . . . . 25.10 Patriot Missile Software Problem . . . . . . . . . . . . . . . . 25.11 Notes and References . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
497 498 499 499 500 501 501 502 504 506 507 508
26 A Gallery of Test Matrices 26.1 The Hilbert and Cauchy Matrices . . . . . . . . . . . . . . . . 26.2 Random Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 26.3 “Randsvd” Matrices . . . . . . . . . . . . . . . . . . . . . . . . 26.4 The Pascal Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 26.5 Tridiagonal Toeplitz Matrices . . . . . . . . . . . . . . . . . . 26.6 Companion Matrices . . . . . . . . . . . . . . . . . . . . . . . 26.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . . 26.7.1 LAPACK . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
513 514 517 519 520 524 525 526 527 527
A Solutions to Problems
529
25.5 25.6 25.7
579 B S i n g u l a r V a l u e D e c o m p o s i t i o n , M-Matrices B.1 Singular Value Decomposition . . . . . . . . . . . . . . . . . . 580 B.2 M-Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 C Acquiring Software C.1 Internet. . . . . . C.2 Netlib . . . . . . . C.3 MATLAB . . . . . C.4 NAG Library and
. . . . . . . . . . . . FTN90
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compiler . . . . . . . . .
D Program Libraries D.1 Basic Linear Algebra Subprograms D.2 EISPACK . . . . . . . . . . . . . . D.3 LINPACK . . . . . . . . . . . . . . D.3 LAPACK . . . . . . . . . . . . . . . D.4.1 Structure of LAPACK . . . E The Test Matrix Toolbox
. . . . .
. . . . .
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. . . . .
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. . . . .
. . . . .
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581 582 582 583 583
. . . . .
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. . . . . . . . . . . . . . . . . . . .
585 586 587 587 587 588
. . . .
. . . .
591
xv Bibliography
595
Name Index
665
Subject Index
675
List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 2.1
4.1 4.2
Backward and forward errors for y = . . . . . . . . . . . . Mixed forward-backward error for y = . . . . . . . . . . . Forward errors and relative residuals ||bversus precision. . . . . . . . . . . . . . . Absolute error versus precision, t = -log2 u . . . . . . . . . . . Relative errors ||Ak -Âk ||2/||A||2 for Givens QR factorization. Values of rational function computed by Horner’s rule. . .
20 21 25 29
Relative distance from to the next larger machine number (β=2, t =24), displaying wobbling precision. . . . . . . . . . .
44
Recovering the rounding error. . . . . . . . . . . . . . . . . . . Errors for Euler’s method with and without compensated summation. . . . . . . . . . . . . . . . . . . . . . . . .
8 9
92 96
5.1
Computed polynomial values and running and a priori bounds for Horner’s method. . . . . . . . . . . . . . . . . . . . . . . . . 107
6.1
Plots of p versus ||A||p, for 1 < p < 15. . . . . . . . . . . . . . . 125
9.1
A banded matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . 182
13.1 Residuals for inverses computed by MATLAB'S INV function. . . 264 14.1 Underestimation ratio for Algorithm 14.4 for 5×5 matrix A( θ). 297 16.1 Forward and backward errors for SOR iteration. 17.1 17.2 17.3 17.4 17.5 17.6
. . . . . . . . 327
A typical hump for a convergent, nonnormal matrix. Diverging powers of a nilpotent matrix, C1 4 . . . . . Infinity norms of powers of 2 × 2 matrix J in (17.2). Computed powers of chebspec matrices. . . . . . . . Pseudospectra for chebspec matrices. . . . . . . . . Pseudospectrum for SOR iteration matrix. . . . . . . xvii
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
347 347 349 356 357 359
XV111
LIST OF FIGURES
. . . . . . . . . . . . . . 363 18.1 Householder matrix P times vector . . . . . . . . . . . . . . . . . . 372 18.2 Givens rotation, y = G(i,j,θ) 22.1 Exponent versus time for matrix multiplication. . . . . . . . . . 449 22.2 Errors for Strassen’s method with two random matrices of dimension n = 1024. . . . . . . . . . . . . . . . . . . . . . . . . . 457 23.1 Error in FFT followed by inverse FFT. . . . . . . . . . . . . . . 468 24.1 The possible steps in one iteration of the MDS method when n=2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 25.1 Rational function r . . . . . . . . . . . . . . . . . . . . . . . . 493 25.2 Error in evaluating rational function r . . . . . . . . . . . . . . 494 26.1 spy(rem(pascal(32),2)). . . . . . . . . . . . . . . . . . . . . 524 26.2 Pseudospectra of compan(A). . . . . . . . . . . . . . . . . , . . 526 26.3 Pseudospectra of 32 × 32 pentadiagonal Toeplitz matrices. . . . 528
List of Tables 1.1 Computed approximations = ƒl((1+1/n) n ) to e = 2.71828 . . . . 16 1.2 Computed values of from Algorithms 1 and 2. . . . 23 1.3 Results from GE without pivoting on an upper Hessenberg mat r i x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.1 2.2 2.3 2.4 2.5
Floating point arithmetic parameters. . . . . . IEEE arithmetic exceptions and default results. Test arithmetics. . . . . . . . . . . . . . . . . . Sine test. . . . . . . . . . . . . . . . . . . . . . Exponentation test. . . . . . . . . . . . . . . .
4.1
Mean square errors for nonnegative
6.1 6.2
Constants αp q such that Constants αpq such that
7.1
Backward and forward stability. . . . . . . . . . . . . . . . . . . 143
9.1 9.2
Times for solution of a linear system of order n . . . . . . . . . 189 Records for largest dense linear systems solved. . . . . . . . . . 199
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41 46 54 55 55
. . . . . . . . . . . . . .
99
. . . . . . . . 121 . . . . . 122
11.1 ω|A|,|b| values for A = orthog(25).
. . . . . . . . . . . . . . . . 240
11.2 ω|A|,|b| values for A = clement(50) . . . . . . . . . . . . . . . . . 241 11.3 ω|A|,|b| values for A = gfpp(50) . . . . . . . . . . . . . . . . . . . 241 12.1 Stability of block and point LU factorization. . . . . . . . . . . 256 13.1 13.2 13.3 13.4 13.5 13.6
Backward errors for the -norm. . . . . . . . . . . . . Mflop rates for inverting a triangular matrix on a Cray 2. . . . Mflop rates for inverting a full matrix on a Cray 2. . . . . . . . Times (minutes and seconds) for inverting an n × n matrix. . . Additional timings for inverting an n × n matrix. . . . . . . . . Gauss-Jordan elimination for U =6. . . . . . . . . . . . . . .
16.1 Dates of publication of selected iterative methods. xix
262 270 275 276 276 279
. . . . . . . 326
LIST OF TABLES 16.2 Result for Jacobi method, α = ½-8 -j . . . . . . . . . . . . . 333 16.3 Results for Jacobi method, a = -(½-8 -j ). . . . . . . . . . . 333 19.1 LS backward errors and residual for Vandermonde system. . . . 405 20.1 Backward errors for underdetermined Vandermonde system. . . 422 21.1 Bounds and estimates for . . . . . . . . . . . . . . . . 428 21.2 Parameters in the three-term recurrence (21.6). . . . . . . . . . 433 21.3 Results for dual Chebyshev-Vandermonde-like system. . . . . . 438 25.1 Results from Cholesky factorization. . . . . . . . . . . . . . . . 496 25.2 Effect of extended run time on Patriot missile operation. . . . . 507 26.1 Condition numbers of Hilbert and Pascal matrices. . . . . . . . 516
Preface It has been 30 years since the publication of Wilkinson’s books Rounding Errors in Algebraic Processes [1088, 1963] and The Algebraic Eigenvalue Problem [1089, 1965]. These books provided the first thorough analysis of the effects of rounding errors on numerical algorithms, and they rapidly became highly influential classics in numerical analysis. Although a number of more recent books have included analysis of rounding errors, none has treated the subject in the same depth as Wilkinson. This book gives a thorough, up-to-date treatment of the behaviour of numerical algorithms in finite precision arithmetic. It combines algorithmic derivations, perturbation theory, and rounding error analysis. Software practicalities are emphasized throughout, with particular reference to LAPACK. The best available error bounds, some of them new, are presented in a unified format with a minimum of jargon. Historical perspective is given to provide insight into the development of the subject, and further information is provided in the many quotations. Perturbation theory is treated in detail, because of its central role in revealing problem sensitivity and providing error bounds. The book is unique in that algorithmic derivations and motivation are given succinctly, and implementation details minimized, so that attention can be concentrated on accuracy and stability results. The book was designed to be a comprehensive reference and contains extensive citations to the research literature. Although the book’s main audience is specialists in numerical analysis, it will be of use to all computational scientists and engineers who are concerned about the accuracy of their results. Much of the book can be understood with only a basic grounding in numerical analysis and linear algebra. This first two chapters are very general. Chapter 1 describes fundamental concepts of finite precision arithmetic, giving many examples for illustration and dispelling some misconceptions. Chapter 2 gives a thorough treatment of floating point arithmetic and may well be the single most useful chapter in the book. In addition to describing models of floating point arithmetic and the IEEE standard, it explains how to exploit “low-level” features not represented in the models and contains a large set of informative exercises. In the rest of the book the focus is, inevitably, on numerical linear algebra, because it is in this area that rounding errors are most influential and have xxi
xxii
PREFACE
been most extensively studied. However, I found that it was impossible to cover the whole of numerical linear algebra in a single volume. The main omission is the area of eigenvalue and singular value computations, which is still the subject of intensive research and requires a book of its own to summarize algorithms, perturbation theory, and error analysis. This book is therefore certainly not a replacement for The Algebraic Eigenvalue Problem. Two reasons why rounding error analysis can be hard to understand are that, first, there is no standard notation and, second, error analyses are often cluttered with re-derivations of standard results. In this book I have used notation that I find nearly always to be the most convenient for error analysis: the key ingredient is the symbol γ n = nu/(1 - nu), explained in §3.1. I have also summarized many basic error analysis results (for example, in Chapters 3 and 8) and made use of them throughout the book. I like to think of these basic results as analogues of the Fortran BLAS (Basic Linear Algebra Subprograms): once available in a standard form they can be used as black boxes and need not be reinvented. A number of the topics included here have not been treated in depth in previous numerical analysis textbooks. These include floating point summation, block LU factorization, condition number estimation. the Sylvester equation, powers of matrices. finite precision behaviour of stationary iterative methods, Vandermonde systems, and fast matrix multiplication, each of which has its own chapter. But there are also some notable omissions. I would have liked to include a chapter on Toeplitz systems, but this is an area in which stability and accuracy are incompletely understood and where knowledge of the underlying applications is required to guide the investigation. The important problems of updating and downdating matrix factorizations when the matrix undergoes a “small” change have also been omitted due to lack of time and space. A further omission is analysis of parallel algorithms for all the problems considered in the book (though blocked and partitioned algorithms and one particular parallel method for triangular systems are treated). Again, there are relatively few results and this is an area of active research. Throughout the history of numerical linear algebra, theoretical advances have gone hand in hand with software development. This tradition has continued with LAPACK (1987-), a project to develop a state-of-the-art Fortran package for solving linear equations and eigenvalue problems. LAPACK has enjoyed a synergy with research that has led to a number of important breakthroughs in the design and analysis of algorithms, from the standpoints of both performance and accuracy. A key feature of this book is that it provides the material needed to understand the numerical properties of many of the algorithms in LAPACK, the except ions being the routines for eigenvalue and singular value problems. In particular, the error bounds computed by the LAPACK linear equation solvers are explained, the LAPACK condition estimator is described in detail, and some of the software issues confronted by
XX111 the LAPACK developers are highlighted. Chapter 25 examines the influence of floating point arithmetic on general numerical software, offering salutary stories, useful techniques, and brief descriptions of relevant codes. This book has been written with numerical analysis courses in mind, although it is not designed specifically as a textbook. It would be a suitable reference for an advanced course (for example, for a graduate course on Numerical Linear Algebra following the syllabus recommended by the ILAS Education Committee [601, 1993]), and could be used by instructors at all levels as a supplementary text from which to draw examples, historical perspective, statements of results, and exercises. The exercises (actually labelled “problems”) are an important part of the book, and many of them have not, to my knowledge, appeared in textbooks before. Where appropriate I have indicated the source of an exercise; a name without a citation means that the exercise came from private communication or unpublished notes. Research problems given at the end of some sets of exercises emphasize that most of the areas covered are still active. In addition to surveying and unifying existing results (including some that have not appeared in the mainstream literature) and sometimes improving upon their presentation or proof, this book contains new results. Some of particular note are as follows. 1. The error analysis in §5.3 for evaluation of the Newton interpolating polynomial. 2. The forward error analysis for iterative refinement in §11.1. 3. The error analysis of Gauss-Jordan elimination in §13.4. 4. The unified componentwise error analysis of QR factorization methods in Chapter 18, and the corresponding analysis of their use for solving the least squares problem in Chapter 19. 5. Theorem 20.3, which shows the backward stability of the QR factorization method for computing the minimum 2-norm solution to an underdetermined system. The Notes and References are an integral part of each chapter. In addition to containing references, historical information, and further details, they include material not covered elsewhere in the chapter, and should always be consulted, in conjunction with the index, to obtain the complete picture. I have included relatively few numerical examples except in the first chapter. There are two reasons. One is to reduce the length of the book. The second reason is because today it so easy for the reader to perform experiments in MATLAB* or some other interactive system. To this end I have made *MATLAB is a registered trademark of The Math Works, Inc.
xxiv
PREFACE
available the Test Matrix Toolbox, which contains MATLAB M-files for many of the algorithms and special matrices described in the book; see Appendix E. This book has been designed to be as easy to use as possible. There are thorough name and subject indexes, page headings show chapter and section titles and numbers, and there is extensive cross-referencing. I have adopted the unusual policy of giving with (nearly) every citation not only its numerical location in the bibliography but also the names of the authors and the year of publication. This provides as much information as possible in a citation and reduces the need for the reader to turn to the bibliography. A database acc-stab-num-alg.bib containing all the references in the bibliography is available over the Internet from the bibnet project (which can be accessed via netlib, described in §C.2). Special care has been taken to minimize the number of typographical and other errors, but no doubt, some remain. I will be happy to receive notification of errors, as well as comments and suggestions for improvement. Acknowledgements Three books, in addition to Wilkinson’s, have strongly influenced my research in numerical linear algebra and have provided inspiration for this book: Golub and Van Loan’s Matrix Computations [470, 1989] (first edition 1983), Parlett’s The Symmetric Eigenvalue Problem [820, 1980], and Stewart’s Introduction to Matrix Computations [941, 1973]. Knuth’s The Art of Computer Programming books [666, 1973-1981] have also influenced my style and presentation. Jim Demmel has contributed greatly to my understanding of the subject of this book and provided valuable technical help and suggestions. The first two chapters owe much to the work of Velvel Kahan; I am grateful to him for giving me access to unpublished notes and for suggesting improvements to early versions of Chapters 2 and 25. Des Higham read various drafts of the book, offering sound advice and finding improvements that had eluded me. Other people who have given valuable help, suggestions, or advice are Zhaojun Bai, Brad Baxter, Åke Björck, Martin Campbell-Kelly, Shivkumar Chandrasekaran, Alan Edelman, Warren Ferguson, Philip Gill, Gene Golub, George Hall, Sven Hammarling, Andrzej Kielbas, Philip Knight, Beresford Parlett, David Silvester, Michael Saunders, Ian Smith, Doron Swade, Nick Trefethen, Jack Williams, and Hongyuan Zha. David Carlisle provided invaluable help and advice concerning . Working with SIAM on the publication of this book was a pleasure. Special thanks go to Nancy Abbott (design), Susan Ciambrano (acquisition), Ed Cilurso (production), Beth Gallagher (copy editing), Corey Gray (production),
XXV Mary Rose Muccie (copy editing and indexing), Colleen Robishaw (design), and Sam Young (production). Research leading to this book has been supported by grants from the Engineering and Physical Sciences Research Council, by a Nuffield Science Research Fellowship from the Nuffield Foundation, and by a NATO Collaborative Research Grant held with J. W. Demmel. I was fortunate to be able to make extensive use of the libraries of the University of Manchester, the University of Dundee, Stanford University, and the University of California, Berkeley. This book was typeset in using the book document style. The references were prepared in and the index with MakeIndex. It is difficult to imagine how I could have written the book without these wonderful tools. I used the “big” software from the distribution, running on a 486DX workstation. I used text editors The Semware Editor (Semware Corporation) and GNU Emacs (Free Software Foundation) and checked spelling with PC-Write (Quicksoft).
Manchester April 1995
Nicholas J. Higham
About the Dedication This book is dedicated to the memory of two remarkable English mathematicians, James Hardy Wilkinson (1919-1986), FRS, and Alan Mathison Turing (1912-1954), FRS, both of whom made immense contributions to scientific computation. Turing’s achievements include his paper “On Computable Numbers, with an Application to the Entscheidungsproblem”, which answered Hilbert’s decidability question using the abstract device now known as a Turing machine [1025, 1936]; his work at Bletchley Park during World War II on breaking the ciphers of the Enigma machine; his 1945 report proposing a design for the Automatic Computing Engine (ACE) at the National Physical Laboratory [1026, 1945]; his 1948 paper on LU factorization and its rounding error analysis [1027, 1948]; his consideration of fundamental questions in artificial intelligence (including his proposal of the “Turing test”); and, during the last part of his life, spent at the University of Manchester, his work on morphogenesis (the development of structure and form in an organism). Turing is remembered through the Turing Award of the Association for Computing Machinery (ACM), which has been awarded yearly since 1966 [3, 1987]. For more about Turing, read the superb biography by Hodges [575, 1983], described by a reviewer as “one of the finest pieces of scholarship to appear in the history of computing” [182, 1984]. Wilkinson, like Turing a Cambridge-trained mathematician, was Turing’s assistant at the National Physical Laboratory. When Turing left, Wilkinson managed the group that built the Pilot ACE, contributing to the design and construction of the machine and its software. Subsequently, he used the machine to develop and study a variety of numerical methods. He developed backward error analysis in the 1950s and 1960s publishing the books Rounding Errors in Algebraic Processes [1088, 1963]† (REAP) and The Algebraic Eigenvalue Problem [1089, 1965]‡ (AEP), both of which rapidly achieved the status of classics. (AEP was reprinted in paperback in 1988 and, after being out of print for many years, REAP is now also available in paperback.) The AEP was described by the late Professor Leslie Fox as “almost certainly the most important and widely read title in numerical analysis”. Wilkinson also † ‡
REAP has been translated into Polish [1091, AEP has been translated into Russian [1094,
xxvii
19 6 7 ] 1970 ].
and German [1093,
19 6 9 ].
XXV111
ABOUT THE DEDICATION
contributed greatly to the development of mathematical software. The volume Handbook for Automatic Computation, Volume II: Linear Algebra [1102, 1971 ], co-edited with Reinsch, contains high-quality, properly documented software and has strongly influenced subsequent software projects such as the NAG Library, EISPACK, LINPACK, and LAPACK. Wilkinson received the 1970 Turing Award. In his Turing Award lecture he described life with Turing at the National Physical Laboratory in the 1940s [1096, 1971]. Wilkinson is remembered through SIAM’s James H. Wilkinson Prize in Numerical Analysis and Scientific Computing, awarded every 4 years; the Wilkinson Prize for Numerical Software, awarded by Argonne National Laboratory, the National Physical Laboratory, and the Numerical Algorithms Group; and the Wilkinson Fellowship in Scientific Computing at Argonne National Laboratory. For more about Wilkinson see the biographical memoir by Fox [403, 1987], Fox’s article [402, 1978], Parlett’s essay [821, 1990], the prologue and epilogue of the proceedings [252, 1990] of a conference held in honour of Wilkinson at the National Physical Laboratory in 1987. and the tributes in [23, 1987]. Lists of Wilkinson’s publications are given in [403, 1987] and in the special volume of the journal Linear Algebra and its Applications (88/89, April 1987) published in his memory.
Previous Home
Chapter 1 Principles of Finite Precision Computation
Numerical precision is the very soul of science. -SIR D’ARCY WENTWORTH THOMPSON, On Growth and Form (1942)
There will always be a small but steady demand for error-analysts to . . . expose bad algorithms’ big errors and, more important, supplant bad algorithms with provably good ones. -WILLIAM M. KAHAN, Interval Arithmetic Options in the Proposed IEEE Floating Point Arithmetic Standard (1980)
Since none of the numbers which we take out from logarithmic and trigonometric tables admit of absolute precision, but are all to a certain extent approximate only, the results of all calculations performed by the aid of these numbers can only be approximately true . . . It may happen, that in special cases the effect of the errors of the tables is so augmented that we may be obliged to reject a method, otherwise the best, and substitute another in its place. -CARL FRIEDRICH GAUSS’, Theoria Motus (1809)
Backward error analysis is no panacea; it may explain errors but not excuse them. -HEWLETT-PACKARD, HP-15C Advanced Functions Handbook (1982)
1
Cited in Goldstine [461,
1977 ,
p. 258].
1
Next
PRINCIPLES OF FINITE PRECISION COMPUTATION
2
This book is concerned with the effects of finite precision arithmetic on numerical algorithms’, particularly those in numerical linear algebra. Central to any understanding of high-level algorithms is an appreciation of the basic concepts of finite precision arithmetic. This opening chapter briskly imparts the necessary background material. Various examples are used for illustration, some of them familiar (such as the quadratic equation) but several less well known. Common misconceptions and myths exposed during the chapter are highlighted towards the end, in §1.19. This chapter has few prerequisites and few assumptions are made about the nature of the finite precision arithmetic (for example, the base, number of digits, or mode of rounding, or even whether it is floating point arithmetic). The second chapter deals in detail with the specifics of floating point arithmetic. A word of warning: some of the examples from §1.12 onward are special ones chosen to illustrate particular phenomena. You may never see in practice the extremes of behaviour shown here. Let the examples show you what can happen, but do not let them destroy your confidence in finite precision arithmetic!
1.1. Notation and Background We describe the notation used in the book and briefly set up definitions needed for this chapter. Generally, we use capital subscripted lower case lower case lower case Greek
A ,B ,C ∆,Λ letters letters αi j , bij, cij , δi j , letters , y, z, c, g, h letters α, β, γ,φ , ρ
ij
for for for for
matrices, matrix elements, vectors, scalars,
following the widely used convention originally introduced by Householder [587, 19 6 4 ]. The vector space of all real m × n matrices is denoted by IR m × n and the vector space of real n-vectors by IRn . Similarly, denotes the vector space of complex m × n matrices. Algorithms are expressed using a pseudocode based on the MATLAB language [232, 1988], [735, 1992]. Comments begin with the % symbol. Submatrices are specified with the colon notation, as used in MATLAB and Fortran 90: A( p :q, r:s) denotes the submatrix of A formed by the intersection of rows p to q and columns r to s. As a special case, a lone colon as the row or column specifier means to take all entries in that row or column; thus A (:,j ) is the jth column of A and A(i,:) the ith row. The values taken by an integer 2
For the purposes of this book an algorithm is a MATLAB program; cf. Smale [924,
1990].
1.1 NOTATION AND BACKGROUND
3
variable are also described using the colon notation: “i = 1:n” means the same as “i = 1,2, . . . , n” . Evaluation of an expression in floating point arithmetic is denoted ƒl(·), and we assume that the basic arithmetic operations op=+,-,*,/ satisfy |δ | < u .
(1.1)
Here, u is the unit roundoff (or machine precision), which is typically of order 10-8 or 10-16 in single and double precision computer arithmetic, respectively, and between 10-10 and 10-12 on pocket calculators. For more on floating point arithmetic see Chapter 2. Computed quantities (and, in this chapter only, arbitrary approximations) wear a hat. Thus denotes the computed approximation to . Definitions are often (but not always) indicated by “:=” or “=:”, with the colon next to the object being defined. We make use of the floor and ceiling functions: is the largest integer less than or equal to , and is the smallest integer greater than or equal to . 2 The normal distribution with mean µ and variance is denoted by We measure the cost of algorithms in flops. A flop is an elementary floating point operation: +,-,/, or *. We normally state only the highest-order terms of flop counts. Thus, when we say that an algorithm for n × n matrices requires 2n 3/3 flops, we really mean 2n 3/3+O(n 2) flops. Other definitions and notation are introduced when needed. Two top ics, however, do not fit comfortably into the main text and are described in Appendix B: the singular value decomposition (SVD) and M-matrices. All our numerical experiments were carried out either in MATLAB 4.2 [735, 1992], sometimes in conjunction with the Symbolic Math Toolbox [204, 1993], or with the Salford Software/Numerical Algorithms Group FTN903 Fortran 90 compiler, Version 1.2 [888, 1993]. Whenever we say a computation was “done in Fortran 90” we are referring to the use of this compiler. All the results quoted were obtained on a 486DX workstation, unless otherwise stated, but many of the experiments were repeated on a Sun SPARCstation, using the NAGWare4 FTN90 compiler [785, 1992]. Both machines use IEEE standard floating point arithmetic and the unit roundoff is u = 2 -53 1.1 × 10 -16 in M ATLAB and in double precision in Fortran 90. (Strictly speaking, in Fortran 90 we should not use the terminology single and double precision but should refer to the appropriate KIND parameters; see, e.g., Metcalf and Reid [749, 1990, §2.6]. However, these terms are vivid and unambiguous in IEEE arithmetic, so we use them throughout the book.) 3 FTN90 is a joint trademark of Salford Software Ltd. and The Numerical Algorithms Group Ltd. 4 NAGWare is a trademark of The Numerical Algorithms Group Ltd.
PRINCIPLES OF FINITE PRECISION COMPUTATION
4
1.2. Relative Error and Significant Digits Let be an approximation to a real number the accuracy of are its absolute error
. The most useful measures of
and its relative error
(which is undefined if = 0). An equivalent definition of relative error is = (1+p). Some authors omit the absolute values = |ρ|, where from these definitions. When the sign is important we will simply talk about “the error . In scientific computation, where answers to problems can vary enormously in magnitude, it is usually the relative error that is of interest, because it is scale independent: scaling and leaves unchanged. Relative error is connected with the notion of correct significant digits (or correct significant figures). The significant digits in a number are the first nonzero digit and all succeeding digits. Thus 1.7320 has five significant digits, while 0.0491 has only three. What is meant by correct significant digits in a number that approximates another seems intuitively clear, but a precise definition is problematic, as we explain in a moment. First, note that for a number with p significant digits there are only p +1 possible answers to the question “how many correct significant digits does have?” (assuming is not a constant such as 2.0 that is known exactly). Therefore the number of correct significant digits is a fairly crude measure of accuracy in comparison with the relative error. For example, in the following two cases agrees with to three but not four significant digits by any reasonable definition, yet the relative errors differ by a factor of about 44: = 1.00000, = 9.00000,
= 1.00499, = 8.99899,
= 4.99 × 10-3, = 1.12 × 10-4.
Here is a possible definition of correct significant digits: an approximation to has p correct significant digits if and round to the same number to p significant digits. Rounding is the act of replacing a given number by the nearest p significant digit number, with some rule for breaking ties when there are two nearest. This definition of correct significant digits is mathematically elegant and agrees with intuition most of the time. But consider the numbers = 0.9949,
= 0.9951.
According to the definition does not have two correct significant digits but does have one and three correct significant digits!
6
PRINCIPLES OF FINITE PRECISION COMPUTATION
or, if the data is itself the solution to another problem, it may be the result of errors in an earlier computation. The effects of errors in the data are generally easier to understand than the effects of rounding errors committed during a computation, because data errors can be analysed using perturbation theory for the problem at hand, while intermediate rounding errors require an analysis specific to the given method. This book contains perturbation theory for most of the problems considered, for example, in Chapters 7 (linear systems), 19 (the least squares problem), and 20 (underdetermined systems). Analysing truncation errors, or discretization errors, is one of the major tasks of the numerical analyst. Many standard numerical methods (for example, the trapezium rule for quadrature, Euler’s method for differential equations, and Newton’s method for nonlinear equations) can be derived by taking finitely many terms of a Taylor series. The terms omitted constitute the truncation error, and for many methods the size of this error depends on a parameter (often called h, “the stepsize”) whose appropriate value is a compromise between obtaining a small error and a fast computation. Because the emphasis of this book is on finite precision computation, with virtually no mention of truncation errors, it would be easy for the reader to gain the impression that the study of numerical methods is dominated by the study of rounding errors. This is certainly not the case. Trefethen explains it well when he discusses how to define numerical analysis [1016, 1992]: Rounding errors and instability are important, and numerical analysts will always be the experts in these subjects and at pains to ensure that the unwary are not tripped up by them. But our central mission is to compute quantities that are typically uncomputable, from an analytic point of view, and to do it with lightning speed. In this quotation “uncomputable” means that approximations are necessary, and thus Trefethen’s point is that developing good approximations is a more fundamental task than analysing the effects of rounding errors on those approximations. A possible way to avoid rounding and truncation errors (but not data errors) is to try to solve a problem using a symbolic manipulation package, such as Maple5 [199, 1991] or Mathematica6 [1109, 1991]. Indeed, we have used this approach to compute “exact answers” in some of our numerical experiments, While we acknowledge the value of symbolic manipulation as part of the toolkit of the scientific problem solver, we do not study it in this book. 5 6
Maple is a registered trademark of Waterloo Maple Software. Mathematica is a registered trademark of Wolfram Research Inc.
1.3 SOURCES OF ERRORS
5
A definition of correct significant digits that does not suffer from the latter anomaly states that agrees with to p significant digits if is less than half a unit in the pth significant digit of . However, this definition implies that 0.123 and 0.127 agree to two significant digits, whereas many people would say that they agree to less than two significant digits. In summary, while the number of correct significant digits provides a useful way in which to think about the accuracy of an approximation, the relative error is a more precise measure (and is base independent). Whenever we give an approximate answer to a problem we should aim to state an estimate or bound for the relative error. When and are vectors the relative error is most often defined with a norm, as For the commonly used norms := maxi := the inequality < ½ × 10-p implies that components with have about p correct significant decimal digits, but for the smaller components the inequality merely bounds the absolute error. A relative error that puts the individual relative errors on an equal footing is the componentwise relative error
which is widely used in error analysis and perturbation analysis (see Chapter 7, for example). As an interesting aside we mention the “tablemaker’s dilemma”. Suppose you are tabulating the values of a transcendental function such as the sine function and a particular entry is evaluated as 0.124|500000000 correct to a few digits in the last place shown, where the vertical bar follows the final significant digit to be tabulated. Should the final significant digit be 4 or 5? The answer depends on whether there is a nonzero trailing digit and, in principle, we may never be able answer the question by computing only a finite number of digits.
1.3. Sources of Errors There are three main sources of errors in numerical computation: rounding, data uncertainty, and truncation. Rounding errors, which are an unavoidable consequence of working in finite precision arithmetic, are largely what this book is about. The remainder of this chapter gives basic insight into rounding errors and their effects. Uncertainty in the data is always a possibility when we are solving practical problems. It may arise in several ways: from errors in measuring physical quantities, from errors in storing the data on the computer (rounding errors),
1.4 PRECISION VERSUS ACCURACY
7
1.4. Precision Versus Accuracy The terms accuracy and precision are often confused or used interchangeably, but it is worth making a distinction between them. Accuracy refers to the absolute or relative error of an approximate quantity. Precision is the accuracy with which the basic arithmetic operations +,-,*,/ are performed, and for floating point arithmetic is measured by the unit roundoff u (see (1.1)). Accuracy and precision are the same for the scalar computation c = a*b, but accuracy can be much worse than precision in the solution of a linear system of equations, for example. It is important to realize that accuracy is not limited by precision, at least in theory. This may seem surprising, and may even appear to contradict many of the results in this book. However, arithmetic of a given precision can be used to simulate arithmetic of arbitrarily high precision, as explained in §25.9. (The catch is that such simulation is too expensive to be of practical use for routine computation.) In all our error analyses there is an implicit assumption that the given arithmetic is not being used to simulate arithmetic of a higher precision.
1.5. Backward and Forward Errors Suppose that an approximation to y = is computed in an arithmetic of precision u, where f is a real scalar function of a real scalar variable. How should we measure the “quality” of ? In most computations we would be happy with a tiny relative error, but this cannot always be achieved. Instead of focusing on the relative error of we can ask “for what set of data have we actually solved our problem?“, that is, for what do we have = ? In general, there may be many such so we should ask for the smallest one. The value of (or min ), possibly divided by , is called the backward error. The absolute and relative errors of are called forward errors, to distinguish them from the backward error. Figure 1.1 illustrates these concepts. The process of bounding the backward error of a computed solution is called backward error analysis, and its motivation is twofold. First, it interprets rounding errors as being equivalent to perturbations in the data. The data frequently contains uncertainties due to previous computations or errors committed in storing numbers on the computer. If the backward error is no larger than these uncertainties then the computed solution can hardly be criticized-it may be the solution we are seeking, for all we know. The second attraction of backward error analysis is that it reduces the question of bounding or estimating the forward error to perturbation theory, which for many problems is well understood (and only has to be developed once, for the
8
PRINCIPLES OF FINITE PRECISION COMPUTATION Input space
Figure 1.1. Backward and forward errors for y = line = computed.
Output space
Solid line = exact; dotted
given problem, and not for each method). We discuss perturbation theory in the next section. A method for computing y = is called backward stable if, for any it produces a computed with a small backward error, that is, = for some small . The definition of “small” will be context dependent. In general, a given problem has several methods of solution, some of which are backward stable and some not. As an example, assumption (1.1) says that the computed result of the operation ± y is the exact result for perturbed data (1 + δ) and y(1 + δ) with |δ| < u; thus addition and subtraction are, by assumption, backward stable operations. Most routines for computing the cosine function do not satisfy = cos( + ) with a relatively small but only the weaker relation + ∆y = cos( + ), with relatively small ∆y and . A result of the form (1.2) is known as a mixed forward-backward error result and is illustrated in Figure 1.2. Provided that and are sufficiently small, (1.2) says that the computed value scarcely differs from the value + that would have been produced by an input + scarcely different from the actual input Even more simply, is almost the right answer for almost the right data. In general, an algorithm is called numerically stable if it is stable in the mixed forward-backward error sense of (1.2) (hence a backward stable algorithm can certainly be called numerically stable). Note that this definition is specific to problems where rounding errors are the dominant form of errors. The term stability has different meanings in other areas of numerical analysis.
1.6 CONDITIONING Input space
Figure 1.2. Mixed forward-backward error for y = line = computed.
Output space
Solid line = exact; dotted
1.6. Conditioning The relationship between forward and backward error for a problem is governed by the conditioning of the problem, that is, the sensitivity of the solution to perturbations in the data. Continuing the y = example of the previous section, let an approximate solution satisfy = Then, assuming for simplicity that f is twice continuously differentiable,
and we can bound or estimate the right-hand side. This expansion leads to the notion of condition number. Since
the quantity
measures, for small the relative change in the output for a given relative change in the input, and it is called the (relative) condition number of f . If or f is a vector then the condition number is defined in a similar way using norms and it measures the maximum relative change, which is attained for some, but not all, vectors As an example, consider the function = log The condition number is = which is large for 1. This means that a small relative change in can produce a much larger relative change in log for 1. The
10
PRINCIPLES OF FINITE PRECISION COMPUTATION
reason is that a small relative change in produces a small absolute change in = log (since and that change in log may be large in a relative sense. When backward error, forward error, and the condition number are defined in a consistent fashion we have the useful rule of thumb that forward error
condition number × backward error,
with approximate equality possible. One way to interpret this rule of thumb is to say that the computed solution to an ill-conditioned problem can have a large forward error. For even if the computed solution has a small backward error, this error can be amplified by a factor as large as the condition number when passing to the forward error. One further definition is useful. If a method produces answers with forward errors of similar magnitude to those produced by a backward stable method, then it is called forward stable. Such a method need not be backward stable itself. Backward stability implies forward stability, but not vice versa. An example of a method that is forward stable but not backward stable is Cramer’s rule for solving a 2 × 2 linear system, which is discussed in §1.10.1.
1.7. Cancellation Cancellation is what happens when two nearly equal numbers are subtracted. It is often, but not always, a bad thing. Consider the function = (1 With = 1.2×10-5 the value of rounded to 10 significant figures is c = 0.9999 9999 99, so that 1 - c = 0.0000 0000 01. -10
Then (1 - c) = 10 /1.44×10-10 = 0.6944. . . , which is clearly wrong given the fact that 0 < < l/2 for all 0. A 10 significant figure approximation to cos is therefore not sufficient to yield a value of with even one correct figure. The problem is that 1 - c has only 1 significant figure. The subtraction 1 - c is exact, but this subtraction produces a result of the same size as the error in c. In other words, the subtraction elevates the importance of the earlier error. In this particular example it is easy to rewrite to avoid the cancellation. Since cos = 1 - 2 sin2
Evaluating this second formula for with a 10 significant figure approximation to sin yields = 0.5, which is correct to 10 significant figures.
1.8 SOLVING A QUADRATIC EQUATION
11
To gain more insight into the cancellation phenomenon consider the subtraction (in exact arithmetic) where and The terms ∆a and ∆b are relative errors or uncertainties in the data, perhaps attributable to previous computations. With x = a - b we have
The relative error bound for is large when |a - b| > y z > 0 then the cancellation in the evaluation of x + (y - z) is harmless.
1.8. Solving a Quadratic Equation Mathematically, the problem of solving the (real) quadratic equation ax 2 bx + c = 0 is trivial: there are two roots (if a 0)) given by
+
(1.3) Numerically, the problem is more challenging, as neither the successful evaluation of (1.3) nor the accuracy of the computed roots can be taken for granted. The easiest issue to deal with is the choice of formula for computing the roots. If b2 >> |4ac| then and so for one choice of sign the formula (1.3) suffers massive cancellation. This is damaging cancellation because one of the arguments, is inexact, so the subtraction brings into prominence the earlier rounding errors. How to avoid the cancellation is well known: obtain the larger root (in absolute value), x 1, from
and the other from the equation x 1 x 2 = c/a.
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PRINCIPLES OF FINITE PRECISION COMPUTATION
Unfortunately, there is a more pernicious source of cancellation: the subtraction b 2 - 4ac. Accuracy is lost here when b 2 4ac (the case of nearly equal roots), and no algebraic rearrangement can avoid the cancellation. The only way to guarantee accurate computed roots is to use double precision (or some trick tantamount to the use of double precision) in the evaluation of b 2 - 4ac. Another potential difficulty is underflow and overflow. If we apply the formula (1.3) in IEEE single precision arithmetic (described in §2.3) to the equation 1020 x2 - 3·1020 x+2·1020 = 0 then overflow occurs, since the maximum floating point number is of order 10 the roots, however, are innocuous: x = 1 and x = 2. Dividing through the equation by max(| a|, |b|, |c |) = 1020 cures the problem, but this strategy is ineffective for the equation 10-20x 2 3x+2·1020 = 0, whose roots are 1020 and 2·1020. In the latter equation we need to scale the variable: defining x = 1020 y gives 1020 y2 -3·10 20 y+ 2 · 1 0 2 0 = 0 , which is the first equation we considered. These ideas can be built into a general scaling strategy (see the Notes and References), but the details are nontrivial. As this discussion indicates, not only is it difficult to devise an accurate and robust algorithm for solving a quadratic equation, but it is a nontrivial task to prepare specifications that define precisely what “accurate” and “robust” mean for a given system of floating point arithmetic.
1.9. Computing the Sample Variance In statistics the sample variance of n numbers x 1, . . . , xn is defined as (1.4) where the sample mean
Computing from this formula requires two passes through the data, one to compute and the other to accumulate the sum of squares. A two-pass computation is undesirable for large data sets or when the sample variance is to be computed as the data is generated. An alternative formula, found in many statistics textbooks, uses about the same number of operations but requires only one pass through the data: (1.5)
1.10 SOLVING LINEAR EQUATIONS
13
This formula is very poor in the presence of rounding errors because it computes the sample variance as the difference of two positive numbers, and therefore can suffer severe cancellation that leaves the computed answer dominated by roundoff. In fact, the computed answer can be negative, an event aptly described by Chan, Golub, and LeVeque [194, 1983] as “a blessing in disguise since this at least alerts the programmer that disastrous cancellation has occurred”. In contrast, the original formula (1.4) always yields a very accurate (and nonnegative) answer, unless n is large (see Problem 1.10). Surprisingly, current calculators from more than one manufacturer (but not Hewlett-Packard) appear to use the one-pass formula, and they list it in their manuals. As an example, if x = [10000, 10001, 10002]T then, in single precision arithmetic (u 6 × 10-8), the sample variance is computed as 1.0 by the two-pass formula (relative error 0) but 0.0 by the one-pass formula (relative error 1). It might be argued that this data should be shifted by some estimate of the mean before applying the one-pass formula which does not change ), but a good estimate is not always available and there are alternative one-pass algorithms that will always produce an acceptably accurate answer. For example, instead of accumulating and we can accumulate
which can be done via the updating formulae (1.6a) (1.6b) after which = Q n / (n - 1). Note that the only subtractions in these recurrences are relatively harmless ones that involve the data xi . For the numerical example above, (1.6) produces the exact answer. The updating formulae (1.6) are numerically stable, though their error bound is not as small as the one for the two-pass formula (it is proportional to the condition number KN in Problem 1.7). The problem of computing the sample variance illustrates well how mathematically equivalent formulae can have different numerical stability properties.
1.10. Solving Linear Equations For an approximate solution y to a linear system Ax = b ( A IRn × n , b n IR ) the forward error is defined as ||x-y||/||x||, for some appropriate norm.
14
PRINCIPLES OF FINITE PRECISION COMPUTATION
Another measure of the quality of y, more or less important depending on the circumstances, is the size of the residual r = b - Ay . When the linear system comes from an interpolation problem, for example, we are probably more interested in how closely Ay represents b than in the accuracy of y. The residual is scale dependent: multiply A and b by a and r is multiplied by a. One way to obtain a scale-independent quantity is to divide by || A|| ||y||, yielding the relative residual
The importance of the relative residual is explained by the following resuit, which was probably first proved by Wilkinson (see the Notes and References). We use the 2-norm, defined by ||x||2 = (xT x)½ and ||A||2 =
Lemma 1.1. With the notation above, and for the 2-norm,
Proof. If (A+∆A)y = b then r := b-Ay = ∆Ay, so ||r| | 2 < | |∆A| | 2 | |y | | 2 , giving (l.7) On the other hand, (A +∆A)y = b for ∆A = ryT /(yT y) and ||∆A||2 = || r||2 /||y||2, so the bound (1.7) is attainable. Lemma 1.1 says that p(y) measures how much A (but not b) must be perturbed in order for y to be the exact solution to the perturbed system, that is, p(y) equals a normwise relative backward error. If the data A and b are uncertain and p(y) is no larger than this uncertainty (e.g., p(y) = O(u)) then the approximate solution y must be regarded as very satisfactory. For other problems the backward error may not be as easy to compute as it is for a general linear system, as we will see for the Sylvester equation (§15.2) and the least squares problem (§19.7). To illustrate these concepts we consider two specific linear equation solvers: Gaussian elimination with partial pivoting (GEPP) and Cramer’s rule.
1.10.1. GEPP Versus Cramer’s Rule Cramer’s rule says that the components of the solution to a linear system Ax = b are given by x i = det(A i (b))/det(A), where Ai ( b) denotes A with its ith column replaced by b. These formulae are a prime example of a method
1.10 SOLVING LINEAR EQUATIONS
15
that is mathematically elegant, but useless for solving practical problems. The two flaws in Cramer’s rule are its computational expense and its numerical instability. The computational expense needs little comment, and is, fortunately, explained in most modern linear algebra textbooks (for example, Strang [961, 1993] cautions the student “it would be crazy to solve equations that way”). The numerical instability is less well known, but not surprising. It is present even for n = 2, as a numerical example shows. We formed a 2 × 2 system Ax = b with condition number K2 (A) = ||A ||2|| A -1||2 1013, and solved the system by both Cramer’s rule and GEPP in MATLAB (unit roundoff u 1.1 × 10 -16 ). The results were as follows, where r = b - A : Cramer’s rule 1.0000 2.0001
1.5075 × 10-7 1.9285 × 10-7
GEPP 1.0002 2.0004
-4.5689 × 10 -17 -2.1931 × 10 -17
The scaled residual for GEPP is pleasantly small-of order the unit roundoff. That for Cramer’s rule is ten orders of magnitude larger, showing that the computed solution from Cramer’s rule does not closely satisfy the equations, or, equivalently, does not solve a nearby system. The solutions themselves are similar, both being accurate to three significant figures in each component but incorrect in the fourth significant figure. This is the accuracy we would expect from GEPP because of the rule of thumb “forward error backward error × condition number”. That Cramer’s rule is as accurate as GEPP in this example, despite its large residual, is perhaps surprising, but it is explained by the fact that Cramer’s rule is forward stable for n = 2; see Problem 1.9. For general n, the accuracy and stability of Cramer’s rule depend on the method used to evaluate the determinants, and satisfactory bounds are not known even for the case where the determinants are evaluated by GEPP. The small residual produced by GEPP in this example is typical: error analysis shows that GEPP is guaranteed to produce a relative residual of order u when n = 2 (see §9.2). To see how remarkable a property this is, consider the rounded version of the exact solution: z = fl(x) = x + ∆x, where ||∆x ||2 1,
which rounds to 1, since the HP 48G works to about 12 decimal digits. Thus for x > 1, the repeated square roots reduce x to 1.0, which the squarings leave unchanged. For 0 < x < 1 we have
on a 12-digit calculator, so we would expect the square root to satisfy
This upper bound rounds to the 12 significant digit number 0.99. . .9. Hence after the 60 square roots we have on the calculator a number x < 0.99. . .9. The 60 squarings are represented by s(x) = and
Because it is smaller than the smallest positive representable number, this result is set to zero on the calculator--a process known as underflow. (The converse situation, in which a result exceeds the largest representable number, is called overflow.) The conclusion is that there is nothing wrong with the calculator. This innocuous-looking calculation simply exhausts the precision and range of a machine with 12 digits of precision and a 3-digit exponent. 1.12.3. An Infinite Sum It is well known that = 1.6449 3406 6848. . . . Suppose we were not aware of this identity and wished to approximate the sum numerically. The most obvious strategy is to evaluate the sum for increasing k until
1.13 INCREASING THE PRECISION
19
the computed sum does not change. In Fortran 90 in single precision this yields the value 1.6447 2532, which is first attained at k = 4096. This agrees with the exact infinite sum to just four significant digits out of a possible nine. The explanation for the poor accuracy is that we are summing the numbers from largest to smallest, and the small numbers are unable to contribute to the sum. For k = 4096 we are forming s + 4096-2 = s + 2-24, where s 1.6. Single precision corresponds to a 24-bit mantissa, so the term we are adding to s “drops off the end” of the computer word, as do all successive terms. The simplest cure for this inaccuracy is to sum in the opposite order: from smallest to largest. Unfortunately, this requires knowledge of how many terms to take before the summation begins. With 109 terms we obtain the computed sum 1.6449 3406, which is correct to eight significant digits. For much more on summation, see Chapter 4.
1.13. Increasing the Precision When the only source of errors is rounding, a common technique for estimating the accuracy of an answer is to recompute it at a higher precision and to see how many digits of the original and the (presumably) more accurate answer agree. We would intuitively expect any desired accuracy to be achievable by computing at a high enough precision. This is certainly the case for algorithms possessing an error bound proportional to the precision, which includes all the algorithms described in the subsequent chapters of this book. However, since an error bound is not necessarily attained, there is no guarantee that a result computed in t digit precision will be more accurate than one computed in s digit precision, for a given t > s; in particular, for a very ill conditioned problem both results could have no correct digits. For illustration, consider the system Ax = b, where A is the inverse of the 5×5 Hilbert matrix and b i = (-l)i i. (For details of the matrices used in this experiment see Chapter 26.) We solved the system in varying precisions with unit roundoffs u = 2 -t , t = 15:40, corresponding to about 4 to 12 decimal places of accuracy. (This was accomplished in MATLAB by using the function chop from the Test Matrix Toolbox to round the result of every arithmetic operation to t bits; see Appendix E.) The algorithm used was Gaussian elimination (without pivoting), which is perfectly stable for this symmetric positive definite matrix. The upper plot of Figure 1.3 shows t against the relative errors and the relative residuals ||b The lower plot of Figure 1.3 gives corresponding results for A = P5 + 5I, where P5 is the Pascal matrix of order 5. The condition numbers (A) are 1.62×102 for the inverse Hilbert matrix and 9.55×105 for the shifted Pascal matrix. In both cases the general trend is that increasing the precision decreases the residual and relative error, but the behaviour is
20
P RINCIPLES
OF
FINITE P RECISION C OMPUTATION
Figure 1.3. Forward errors and relative residuals ||b versus precision t = - log2 u on the x axis. not monotonic. The reason for the pronounced oscillating behaviour of the relative error (but not the residual) for the inverse Hilbert matrix is not clear. An example in which increasing the precision by several bits does not improve the accuracy is the evaluation of y = x + a sin(bx),
a = 10-8,
b = 224.
(1.8)
Figure 1.4 plots t versus the absolute error, for precisions u = 2-t, t = 10:40. Since a sin(bx) -8.55×10-9, for t less than about 20 the error is dominated by the error in representing x = l/7. For 22 < t < 31 the accuracy is (exactly) constant! The plateau over the range 22 < t < 31 is caused by a fortuitous rounding error in the addition: in the binary representation of the exact answer the 23rd to 32nd digits are 1s, and in the range of t of interest the final rounding produces a number with a 1 in the 22nd bit and zeros beyond, yielding an unexpectedly small error that affects only bits 33rd onwards. A more contrived example in which increasing the precision has no beneficial effect on the accuracy is the following evaluation of z = f(x): y = abs(3(x-0.5)-0.5)/25 if y = 0 z = l
1.14 C ANCELLATION
OF
R OUNDING E RRORS
21
Figure 1.4. Absolute error versus precision, t = -log2 u. else z = ey % Store to inhibit extended precision evaluation. z = (z - 1)/y end In exact arithmetic, z = f(2/3) = 1, but in Fortran 90 on a Sun SPARCstation and on a 486DX workstation, = fl(f(2/3)) = 0.0 in both single and double precision arithmetic. A further example is provided by the “innocuous calculation” of §1.12.2, in which a step function is computed in place of f(x) = x for a wide range of precisions. It is worth stressing that how precision is increased can greatly affect the results obtained. Increasing the precision without preserving important properties such as monotonicity of rounding can vitiate an otherwise reliable algorithm. Increasing the precision without maintaining a correct relationship among the precisions in different parts of an algorithm can also be harmful to the accuracy.
1.14. Cancellation of Rounding Errors It is not unusual for rounding errors to cancel in stable algorithms, with the result that the final computed answer is much more accurate than the inter-
P RINCIPLES
22
OF
FINITE P RECISION C OMPUTATION
mediate quantities. This phenomenon is not universally appreciated, perhaps because we tend to look at the intermediate numbers in an algorithm only when something is wrong, not when the computed answer is sat isfactory. We describe two examples. The first is a very short and rather unusual computation, while the second involves a well-known algorithm for computing a standard matrix decomposition. 1.14.1. Computing (ex
-
1)/x
Consider the function f(x) = (ex - 1)/x = which arises in various applications. The obvious way to evaluate f is via the algorithm % Algorithm 1. if x = 0 f = l else f = (ex - 1)/x end This algorithm suffers severe cancellation for |x | ,
At the end of an analysis it is necessary to bound | - 1|, for which any of the techniques described above can be used. Wilkinson explained in [1100, 1985] that he used a similar notation in his research, writing for a product of r factors 1 + δi with |δi | < u. He also derived results for specific values of n, before treating the general case-a useful trick of the trade! An alternative notation to fl(·) to denote the rounded value of a number or the computed value of an expression is [·], suggested by Kahan. Thus, we would write [a + [b * c]] instead of fl(a + fl(b * c)).
BASICS
76
A completely different notation for error analysis has been proposed by Olver [807, 1978], and subsequently used by him and several other authors. For scalars x and y of the same sign, Olver defines the relative precision rp as follows: y x; rp(a) means that y = eδ x, | δ| < a. Since eδ = 1 + δ + O(δ2), this definition implies that the relative error in x as an approximation to y (or vice versa) is at most a + O (a 2). But, unlike the usual definition of relative error, the relative precision possesses the properties of symmetry: y x ; rp( a) x x; rp(a ) and z y; rp(β) additivity: y
y; rp(a), z
x : rp(a + β).
Proponents of relative precision claim that the symmetry and additivity properties make it easier to work with than the relative error. Pryce [845, 1981] gives an excellent appraisal of relative precision, with many examples. He uses the additional notation 1(δ ) to mean a number θ with θ 1; rp(δ). The 1(δ) notation is the analogue for relative precision of Stewart’s counter for relative error. In later papers, Pryce extends the rp notation to vector and matrices and shows how it can be used in the error analysis of some basic matrix computations [846, 1984], [847, 1985]. Relative precision has not achieved wide use. The important thing for an error analyst is to settle on a comfortable notation that does not hinder the thinking process. It does not really matter which of the notations described in this section is used, as long as the final result is informative and expressed in a readable form.
3.5. Matrix Multiplication Given error analysis for inner products it is straightforward to analyse matrixvector and matrix-matrix products. Let A IR m × n, x IR n and y = Ax. The vector y can be formed as m inner products, yi = = 1:m, where is the ith row of A. From (3.4) we have
This gives the backward error result (3.10) which implies the forward error bound (3.11)
3.5 M ATRIX M ULTIPLICATION
77
Normwise bounds readily follow (see Chapter 6 for norm definitions): for example, This inner product formation of y can be expressed algorithmically as % Sdot or inner product form. y(1:m) = 0 for i = 1:m for j = 1:n e n d y ( i ) =y ( i ) +a ( i , j) x ( j ) end The two loops can be interchanged to give % Saxpy form. y(1:m) = 0 for j = 1:n for i = 1:m y (i) = y(i) +a( i,j ) x ( j ) end end The terms “sdot” and “saxpy” come from the BLAS (see §D.1). Sdot stands for (single precision) dot product, and saxpy for (single precision) a times x plus y. The question of interest is whether (3.10) and (3.11) hold for the saxpy form. They do: the saxpy algorithm still forms the inner products but instead of forming each one in turn it evaluates them all “in parallel”, a term at a time. The key observation is that exactly the same operations are performed, and hence exactly the same rounding errors are committed-the only difference is the order in which the rounding errors are created. This “rounding error equivalence” of algorithms that are mathematically identical but algorithmically different is one that occurs frequently in matrix computations. The equivalence is not always as easy to see as it is for matrixvector products. Now consider matrix multiplication: C = AB, where A IRm×n and n×p B IR . Matrix multiplication is a triple loop procedure, with six possible loop orderings, one of which is C(1:m,1:p) = 0 for i = 1:m for j = 1:p for k = 1:n C ( i , j ) =C ( i , j ) +A ( i , k ) B ( k , j )
78
BASICS end end end
As for the matrix -vector product. all six versions commit the same rounding errors, so it suffices to consider any one of them. The “jik” and “jki” orderings both compute C a column at a time: c j = Ab j, where cj = C(:,j) and b j = B(:,j). From (3.10),
Hence the jth computed column of C has a small backward error: it is the exact jth column for slightly perturbed data. The same cannot be said for as a whole (see Problem 3.5 for a possibly large backward error bound). However, we have the forward error bound (3.12) and the corresponding normwise bounds include
The bound (3.12) falls short of the ideal bound which says that each component of C is computed with high relative accuracy. Nevertheless (3.12) is the best bound we can expect, because it reflects the sensitivity of the product to componentwise relative perturbations in the data: for any i and j we can find a perturbation ∆A with |∆A| < u|A| such that |(A+∆A)B-AB|ij =u(|A||B|)ij (similarly for perturbations in B).
3.6. Complex Arithmetic To carry out error analysis of algorithms in complex arithmetic we need a model for the basic arithmetic operations. Since complex arithmetic must be implemented using real arithmetic, the complex model is a consequence of the corresponding real one. We will assume that for complex numbers x = a + ib and y = c + id we compute x ± y = a + c ± i(b + d), xy = ac - bd + i(ad + bc),
(3.13a) (3.13b) (3.13c)
3.6 COMPLEX ARITHMETIC
79
Lemma 3.5. For x, y the basic arithmetic operations computed according to (3.13) under the standard model (2.4) satisfy
Proof. Throughout the proof, δi denotes a number bounded by |δi | < u. Addition/subtraction:
so as required. Multiplication:
where
as required. Division:
Then
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B ASICS
where, using Lemma 3.3,
Using the analogous formula for the error in fl(Imx / y ) ,
which completes the proof. It is worth stressing that δ in Lemma 3.5 is a complex number, so we cannot conclude from the lemma that the real and imaginary parts of fl( x op y) are obtained to high relative accuracy---only that they are obtained to high accuracy relative to |x op y|. As explained in §25.8, the formula (3.13c) is not recommended for practical use since it is susceptible to overflow. For the alternative formula (25.1), which avoids overflow, similar analysis to that in the proof of Lemma 3.5 shows that
Bounds for the rounding errors in the basic complex arithmetic operations are rarely given in the literature. Indeed, virtually all published error analyses in matrix computations are for real arithmetic. However, because the bounds of Lemma 3.5 are of the same form as for the standard model (2.4) for real arithmetic, most results for real arithmetic (including virtually all those in this book) are valid for complex arithmetic, provided that the constants are increased appropriately.
3.7. Miscellany In this section we give some miscellaneous results that will be needed in later chapters. The first two results provide convenient ways to bound the effect of perturbations in a matrix product. The first result uses norms and the second. components. Lemma 3.6. If X j + ∆ X consistent norm, then
j
IRn×n satisfies ||∆Xj|| < dj||Xj|| for all j for a
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3.7 MISCELLANY
Proof. The proof is a straightforward induction, which we leave as an exercise (Problem 3.10). A componentwise result is entirely analogous. Lemma 3.7. If Xj + ∆Xj
IRn×n satisfies |∆X j| < δj |X j | for all j, then
The final result describes the computation of the “rank 1 update” y = (I - abT )x, which is an operation arising in various algorithms, including the Gram-Schmidt method and Householder QR factorization. Lemma 3.8. Let a, b, x IR n and let y = (I - ab T )x be computed as fl(x- a( b T x )). Then = y + ∆y, where
so that Proof. Consider first the computation of w = a(bTx). We have
where Finally,
= fl(x -
) satisfies
and
Hence
= y + ∆y, where
=
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B ASICS
3.8. Error Analysis Demystified The principles underlying an error analysis can easily be obscured by the details. It is therefore instructive to examine the basic mechanism of forward and backward error analyses. We outline a general framework that reveals the essential simplicity. Consider the problem of computing z = f(a), where f : IRn IRm . Any algorithm for computing z can be expressed as follows. Let x 1 = a and x k+l = g k (x k), k = 1:p, where
The kth stage of the algorithm represents a single floating point operation and xk contains the original data together with all the intermediate quantities computed so far. Finally, z = where is comprised of a subset of the columns of the identity matrix (so that each zi is a component of xp+1). In floating point arithmetic we have
where ∆xk+i represents the rounding errors on the kth stage and should be easy to bound. We assume that the functions g k are continuously differentiable and denote the Jacobian of gk at a by Jk. Then, to first order,
The pattern is clear: for the final
we have
In a forward error analysis we bound f(a) - , which requires bounds for (products of) the Jacobians Jk. In a backward error analysis we write, again to first order,
3.9 O THER A PPROACHES
83
where Jf is the Jacobian of f. So we need to solve, for ∆a,
In most matrix problems there are fewer outputs than inputs (m < n), so this is an underdetermined system. For a normwise backward error analysis we want a solution of minimum norm. For a componentwise backward error analysis, in which we may want (for example) to minimize subject to | ∆ a| < we can write
and then we want the solution c of minimal m-norm. The conclusions are that forward error analysis corresponds to bounding derivatives and that backward error analysis corresponds to solving a large underdetermined linear system for a solution of minimal norm. In principal, therefore, error analysis is straightforward! Complicating factors in practice are that the Jacobians Jk may be difficult to obtain explicitly, that an error bound has to be expressed in a form that can easily be interpreted, and that we may want to keep track of higher-order terms.
3.9. Other Approaches In this book we do not describe all possible approaches to error analysis. Some others are mentioned in this section. Linearized rounding error bounds can be developed by applying equations that describe, to first order, the propagation of absolute or relative errors in the elementary operations +,-,*,/. The basics of this approach are given in many textbooks (see, for example, Dahlquist and Björck [262, 1974, §2.2] or Stoer and Bulirsch [955, 1980, §1.3]), but for a thorough treatment see Stummel [963, 198 0], [964, 198 1]. Ziv [1133, 1995] shows that linearized bounds can be turned into true bounds by increasing them by a factor that depends on the algorithm. Rounding error analysis can be phrased in terms of graphs. This appears to have been first suggested by McCracken and Dorn [743, 196 4], who use “process graphs” to represent a numerical computation and thereby to analyse the propagation of rounding errors. Subsequent more detailed treatments include those of Bauer [82, 1974], Miller [758, 1976], and Yalamov [1117, 1994]. The work on graphs falls under the heading of automatic error analysis (for more on which see Chapter 24) because processing of the large graphs required to represent practical computations is impractical by hand. Linnainmaa [705, 1976] shows how to compute the Taylor series expansion of the forward error
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B ASICS
in an algorithm in terms of the individual rounding errors, and he presents a graph framework for the computation. Some authors have taken a computational complexity approach to error analysis, by aiming to minimize the number of rounding error terms in a forward error bound, perhaps by rearranging a computation. Because this approach ignores the possibility of cancellation of rounding errors, the results need to be interpreted with care. See Aggarwal and Burgmeier [6, 1979] and Tsao [1024, 1983].
3.10. Notes and References The use of Lemma 3.1 for rounding error analysis appears to originate with the original 1972 German edition of a book by Stoer and Bulirsch [955, 1980]. The lemma is also used, with p i = 1, by Shampine and Allen [911, 1973, p. 18]. Lemma 3.4 is given by Forsythe and Moler [396, 1967, p. 92]. Wilkinson made frequent use of a slightly different version of Lemma 3.4 in which the assumption is nu < 0.1 and the bound for |ηn | is 1.06nu (see, e.g., [1089, 1965, p. 113]). A straight forward notation for rounding errors that is subsumed by the notation described in this chapter is suggested by Scherer and Zeller [899, 1980]. Ziv [1131, 1982] proposes the relative error measure
for vectors x and y and explains some of its favourable properties for error analysis. Wilkinson [1089, 1965, p. 447] gives error bounds for complex arithmetic; Olver [808, 198 3] does the same in the relative precision framework. Demmel [280, 1984] gives error bounds that extend those in Lemma 3.5 by taking into account the possibility of underflow. Henrici [521, 1980] gives a brief, easy to read introduction to the use of the model (2.4) for analysing the propagation of rounding errors in a general algorithm. He uses a set notation that is another possible notation to add to those in §3.4. The perspective on error analysis in §3.8 was suggested by J. W. Demmel.
Problems 3.1. Prove Lemma 3.1. 3.2. (Kielbasiniski and Schwetlick [658, 1988], [659, 1992]) Show that if pi 1 in Lemma 3.1 then the stronger bound |φ n | < nu/(1-½nu) holds for nu < 2.
P ROBLEMS
85
3.3. One algorithm for evaluating a continued fraction
is q n+1 = an +1 for k = n:-1:0 qk = ak + bk/qk+ end
1
Derive a running error bound for this algorithm. 3.4. Prove Lemma 3.3. 3.5. (Backward error result for matrix multiplication.) Let A IRn×n and n×n B IR both be nonsingular. Show that fl(AB) = (A + ∆A)B, where | ∆A| < γn |A||B||B -1|, and derive a corresponding bound in which B is perturbed. 3.6. (Backward error definition for matrix multiplication.) Let A IRm × n n×p and B IR be of full rank and suppose C AB. Define the componentwise backward error
where E and F have nonnegative entries. Show that
where R = C - AB and G = EF. Explain why the definition of w makes sense only when A and B have full rank. Define a mixed backward/forward error applicable to the general case. 3.7. Give analogues of the backward error results (3.4) and (3.10) for complex x, y, and A. 3.8. Let Al,. . . ,Ak
IRn × n . Show that
3.9. Which is the more accurate way to compute x2 - y2: as x2 - y2 or as (x+y)(x-y)?(Assume the use of a guard digit. Note that this computation arises when squaring a complex number.)
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BASICS
3.10. Prove Lemma 3.6. 3.11. (Kahan [629, 1980]) Consider this MATLAB function, which returns the absolute value of its first argument x IRn : function z = absolute(x , m ) y = x.-2; for i=l:m y = sqrt(y); end z = y; for i=l:m-1 z = z. -2; end Here is some output from a 486DX workstation that uses IEEE standard double precision arithmetic: > > x = [ . 2 5 . 5 . 7 5 1 . 2 5 1 . 5 2 ] ; z = a b s o l u t e ( x , 5 0 ) ; [ x ; z] ans = 0.2500 0.5000 0.7500 1.2500 1.5000 2.0000 0.2528 0.5028 0.7788 1.2840 1.4550 2.1170 Give an error analysis to explain the results. The same machine produced this output: > > x = [ . 2 5 . 5 . 7 5 1 . 2 5 1 . 5 2 ] ; z = a b s o l u t e ( x , 7 5 ) ; [ x ; z] ans = 0.2500 0.5000 0.7500 1.2500 1.5000 2.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 But a Sun SPARCstation, which also uses IEEE standard double precision arithmetic, produced ans = 0.2500 0.5000 0.7500 1.2500 1.5000 2.0000 0 0 0 1.0000 1.0000 1.0000 Explain these results and why they differ (cf. §1.12.2). 3.12. Consider the quadrature rule
where the weights wi and nodes xi are assumed to be floating point numbers. Assuming that the sum is evaluated in left-to-right order and that
obtain and interpret a bound for
where
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Chapter 4
Summation
I do hate sums. There is no greater mistake than to call arithmetic an exact science. There are . . . hidden laws of Number which it requires a mind like mine to perceive. For instance, if you add a sum from the bottom up, and then again from the top down, the result is always different. -MRS. LA TOUCHE10
Joseph Fourier introduced this delimited z-notation in 1820, and it soon took the mathematical world by storm. -RONALD L. GRAHAM, DONALD E. KNUTH, and OREN PATASHNIK, Concrete Mathematics (1989)
One of the major difficulties in a practical [error] analysis is that of description. An ounce of analysis follows a pound of preparation. -BERESFORD N. PARLETT, Matrix Eigenvalue Problems (1965)
10
Quoted in Mathematical Gazette [730,
1924 ].
87
88
SUMMATION
Sums of floating point numbers are ubiquitous in scientific computing. They occur when evaluating inner products, means, variances, norms, and all kinds of nonlinear functions. Although at first sight summation might appear to offer little scope for algorithmic ingenuity, the usual “recursive summation” (with various orderings) is just one of a variety of possible techniques. We describe several summation methods and their error analyses in this chap ter. No one method is uniformly more accurate than the others, but some guidelines can be given on the choice of method in particular cases.
4.1. Summation Methods In most circumstances in scientific computing we would naturally translate a sum into code of the form s =0 for i = 1:n s=s+ x end
i
This is known as recursive summation. Since the individual rounding errors depend on the operands being summed, the accuracy of the computed sum varies with the ordering of the xi . (Hence Mrs. La Touche, quoted at the beginning of the chapter, was correct if we interpret her remarks as applying to floating point arithmetic.) Two interesting choices of ordering are the increasing order |x1| < | x 2 | < . . . < |x n |, and the decreasing order |x 1| > |x2| > . . . > |xn|. Another method is pairwise summation (also known as cascade summation, or fan-in summation), in which the xi are summed in pairs according to and this pairwise summation process is repeated recursively on the yi, i = 1 : [ ( n+1)/2]. The sum is obtained in [log2 n] stages. For n = 6, for example, pairwise summation forms S 6 = (( x 1 + x 2 ) + ( x 3 + x 4 ) ) + ( x 5 + x 6 ) . Pairwise summation is attractive for parallel computing, because each of the [log2 n] stages can be done in parallel [573, 1988, §5.2.2]. A third summation method is the insertion met hod. First, the x i are sorted by order of increasing magnitude (alternatively, some other ordering could be used). Then x1 + x2 is formed, and the sum is inserted into the l i s t x2,. . . ,x n , maintaining the increasing order. The process is repeated recursively until the final sum is obtained. In particular cases the insertion
4.2 ERROR ANALYSIS
89
method reduces to one of the other two. For example, if x i = 2 i -1, the insertion method is equivalent to recursive summation, since the insertion is always to the bottom of the list: 1248
348
78
15.
On the other hand, if 1 < x1 < x2 < ... < xn < 2, every insertion is to the end of the list, and the method is equivalent to pairwise summation if n is a power of 2; for example, if 0 < < 1/2,
To choose between these met hods we need error analysis, which we develop in the next section.
4.2. Error Analysis Error analysis can be done individually for the recursive, pairwise and insertion summation methods, but it is more profitable to recognize that each is a special case of a general algorithm and to analyse that algorithm. Algorithm 4.1. Given numbers x1,. . . ,xn this algorithm computes Sn
=
L e t S = {x 1,...,xn }. repeat while S contains more than one element Remove two numbers x and y from S, and add their sum x + y to S. end Assign the remaining element of S to S n . Note that since there are n numbers to be added and hence n - 1 additions to be performed, there must be precisely n - 1 executions of the repeat loop. First, let us check that the previous methods are special cases of Algorithm 4.1. Recursive summation (with any desired ordering) is obtained by taking x at each stage to be the sum computed on the previous stage of the algorithm. Pairwise summation is obtained by [log2 n] groups of executions of the repeat loop, in each group of which the members of S are broken into pairs, each of which is summed. Finally, the insertion method is, by definition, a special case of Algorithm 4.1. Now for the error analysis. Express the ith execution of the repeat loop as Ti = xi1 + yil. The computed sums satisfy (using (2.5)) (4.1)
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SUMMATION
The local error introduced in forming The overall error is the sum of the local errors (since summation is a linear process), so overall we have
(4.2) The smallest possible error bound is therefore
(4.3) (This is actually in the form of a running error bound, because it contains the computed quantities-see §3.3.) It is easy to see that for each i, and so we have also the weaker bound (4.4) This is a forward error bound. A backward error result showing that is the exact sum of terms with can be deduced from (4.1), using the fact that no number xi takes part in more than n - 1 additions. The following criterion is apparent from (4.2) and (-1.3): In designing or choosing a summation met hod to achieve high accuracy, the aim should be to minimize the absolute values of the intermediate sums Ti . The aim specified in this criterion is surprisingly simple to state. When we consider specific methods, however, we find that the aim is difficult to achieve. Consider recursive summation, for which Ideally, we would like to choose the ordering of the xi to minimize This is a combinatorial optimization problem that is too expensive to solve in the context of summation. A reasonable compromise is to determine the ordering sequentially by minimizing, in turn, |x 1|, |S 2|, . . . , |S n-1|. This ordering strategy, which we denote by Psum, can be implemented with O(n log n) comparisons. If we are willing to give up the property that the ordering is influenced by the signs of the xi we can instead use the increasing ordering, which in general will lead to a larger value of than that for the Psum ordering. If all the x i have the same sign then all these orderings are equivalent. Therefore when summing nonnegative numbers by recursive summation the increasing ordering is the best ordering, in the sense of having the smallest a priori forward error bound.
4.2 ERROR ANALYSIS
91
How does the decreasing ordering fit into the picture? For the summation of positive numbers this ordering has little to recommend it. The bound (4.3) is no smaller, and potentially much larger, than it is for the increasing ordering. Furthermore, in a sum of positive terms that vary widely in magnitude the decreasing ordering may not allow the smaller terms to contribute to the sum (which is why the harmonic sum “converges” in floating point arithmetic as n ). However, consider the example with n = 4 and x = [ l , M,
2M,
-3M],
(4.5) where M is a floating point number so large that fl (1 + M) = M (thus M > u -1). The three orderings considered so far produce the following results: Increasing:
= fl(1 + M + 2M - 3M) = 0,
Psum:
= fl(1 + M - 3M + 2M) = 0,
Decreasing:
= fl(-3M + 2M + M + 1) = 1.
Thus the decreasing ordering sustains no rounding errors and produces the exact answer, while both the increasing and Psum orderings yield computed sums with relative error 1. The reason why the decreasing ordering performs so well in this example is that it adds the “1” after the inevitable heavy cancellation has taken place, rather than before, and so retains the important information in this term. If we evaluate the term in the error bound (4.3) for example (4.5) we find Increasing: µ = 4M,
Psum: µ = 3M, Decreasing: µ = M + 1,
so (4.3) “predicts” that the decreasing ordering will produce the most accurate answer, but the bound it provides is extremely pessimistic since there are no rounding errors in this instance. Extrapolating from this example, we conclude that the decreasing ordering is likely to yield greater accuracy than the increasing or Psum orderings whenever there is heavy cancellation in the sum, that is, whenever Turning to the insertion method, a good explanation of the insertion strategy is that it attempts to minimize, one at a time, the terms in the error bound (4.3). Indeed, if the xi are all nonnegative the insertion method minimizes this bound over all instances of Algorithm 4.1. Finally, we note that a stronger form of the bound (4.4) holds for pairwise summation. It can be deduced from (4.3) or derived directly, as follows. Assume for simplicity that n = 2 r . Unlike in recursive summation each addend takes part in the same number of additions, log2 n. Therefore we have a relation of the form
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SUMMATION
Figure 4.1. Recovering the rounding error.
which leads to the bound (4.6) Since it is proportional to log2 n rat her than n, this is a smaller bound than (4.4), which is the best bound of this form that holds in general for Algorithm 4.1.
4.3. Compensated Summation We have left to last the compensated summation method, which is recursive summation with a correction term cleverly designed to diminish the rounding errors. Compensated summation is worth considering whenever an accurate sum is required and computations are already taking place at the highest precision supported by the hardware or the programming language in use. In 1951 Gill [449, 1951] noticed that the rounding error in the sum of two numbers could be estimated by subtracting one of the numbers from the sum, and he made use of this estimate in a Runge-Kutta code in a program library for the EDSAC computer. Gill’s estimate is valid for fixed point arithmetic only. Kahan [625, 196 5] and Møller [777, 196 5] both extended the idea to floating point arithmetic. Møller shows how to estimate a + b - fl (a + b) in chopped arithmetic, while Kahan uses a slightly simpler estimate to derive the compensated summation method for computing The estimate used by Kahan is perhaps best explained with the aid of a diagram. Let a and b be floating point numbers with |a| > |b|, let = fl(a+b), and consider Figure 4.1, which uses boxes to represent the mantissas of a and b. The figure suggests that if we evaluate
4.3 C OMPENSATED S UMMATION
93
in floating point arithmetic, in the order indicated by the parentheses, then the computed ê will be a good estimate of the error (a + b) In fact, for rounded floating point arithmetic in base 2, we have a+ b =
+ê,
(4.7)
that is, the computed ê represents the error exactly. This result (which does not hold for all bases) is proved by Dekker [275, 1971, Thm. 4.7], Knuth [668, 1981, Thm. C, p. 221], and Linnainmaa [703, 1974, Thm. 3]. Note that there is no point in computing fl( + ê), since is already the best floating point representation of a + b ! Kahan’s compensated summation method employs the correction e on every step of a recursive summation. After each partial sum is formed, the correction is computed and immediately added to the next term xi before that term is added to the partial sum. Thus the idea is to capture the rounding errors and feed them back into the summation. The method may be written as follows. Algorithm 4.2 (compensated summation). Given floating point numbers x1, . . . , xn this algorithm forms the sum by compensated summation. s=0;e= 0 for i = 1:n temp = s y=xi+e s=temp+y e = (temp - s) + y % Evaluate in the order shown. end The compensated summation method has two weaknesses: ê is not necessarily the exact correction, since (4.7) is based on the assumption that | a | > |b|, and the addition y = xi + e is not performed exactly. Nevertheless, the use of the corrections brings a benefit in the form of an improved error bound. Knuth [668, 1981, Ex. 19, pp. 229, 572-573] shows that the computed sum satisfies (4.8) which is an almost ideal backward error result (a more detailed version of Knuth’s proof is given by Goldberg [457, 1991]). In [627, 1972] and [628, 1973] Kahan describes a variation of compensated summation in which the final sum is also corrected (thus “s = s + e” is appended to the algorithm above). Kahan states in [627, 1972] and proves in
94
SUMMATION
[628, 1973 ] that (4.8) holds with the stronger bound |µ i | < 2u+O( ( n-i+ 1 ) u 2) The proofs of (4.8) given by Knuth and Kahan are similar; they use the model (2.4) with a subtle induct ion and some intricate algebraic manipulation. The forward error bound corresponding to (4.8) is
(4.9) As long as nu < 1, the constant in this bound is independent of n, and so the bound is a significant improvement over the bounds (4.4) for recursive summation and (4.6) for pairwise summation. Note, however, that if compensated summation is not guaranteed to yield a small relative error. Another version of compensated summation has been investigated by several authors: Jankowski, Smoktunowicz, and Wozniakowski [609, 1983], Jankowski and Wozniakowski [611, 1985], Kielbasinski [655, 1973], Neumaier [788, 1974], and Nickel [797, 1970]. Here, instead of immediately feeding each correction back into the summation, the correct ions are accumulated separately by recursive summation and then the global correction is added to the computed sum. For this version of compensated summation Kielbasinski [655, 1973] and Neumaier [788, 1974] show that (4.10) provided nu < 0.1; this is weaker than (4.8) in that the second-order term has an extra factor n. If n 2 u < 0.1 then in (4.10), |µ i | < 2.1u. Jankowski, Smoktunowicz, and Wozniakowski [609, 198 3] show that, by using a divide and conquer implementation of compensated summation, the range of n for which |µ i | < cu holds in (4.10) can be extended, at the cost of a slight increase in the size of the constant c. Neither the correction formula (4.7) nor the result (4.8) for compensated summation holds under the no-guard-digit model of floating point arithmetic. Indeed, Kahan [634, 1990] constructs an example where compensated summation fails to achieve (4.9) on certain Cray machines, but he states that such failure is extremely rare. In [627, 1972] and [628, 1973] Kahan gives a modification of the compensated summation algorithm in which the assignment “e = (temp - s) + y” is replaced by f = 0 if sign(temp) = sign(y), f = (0.46 * s - s) + s, end e = ( ( temp - f ) - ( s - f )) + y
4.3 C OMPENSATED S UMMATION
95
Kahan shows in [628, 1973] that the modified algorithm achieves (4.8) “on all North American machines with floating hardware” and explains that “The mysterious constant 0.46, which could perhaps be any number between 0.25 and 0.50, and the fact that the proof requires a considerat ion of known machines designs, indicate that this algorithm is not an advance in computer science.” Viten’ko [1056, 1968] shows that under the no-guard-digit model (2.6) the summation met hod with the optimal error bound (in a certain sense defined in [1056, 1968]) is pairwise summation. This does not contradict Kahan’s result because Kahan uses properties of the floating point arithmetic beyond those in the no-guard-digit model. A good illustration of the benefits of compensated summation is provided by Euler’s method for the ordinary differential equation initial value problem y' = f(x,y), y(a) given, which generates an approximate solution according to y k+l = yk + hfk , y0 = y(a). We solved the equation y' = -y with y(0) = 1 over [0,1] using n steps of Euler’s method (nh = 1), with n ranging from 10 to 108. With compensated summation we replace the statements x = x + h, y = y + h * f(x,y) by (with the initialization cx = 0, cy = 0) dx = h + cx new-x = x + dx cx = (x - new-x) + dx x = new_x dy = h * f(x,y) + cy new-y = y + dy cy = (y - new-y) + dy y = new-y Figure 4.2 shows the errors en = |y (1) where is the computed approximation to y(1). The computations were done in Fortran 90 in single precision arithmetic on a Sun SPARCstation (u 6 × 10-8). Since Euler’s method has global error of order h, the error curve on the plot should be approximately a straight line. For the standard. implement at ion of Euler’s method the errors en start to increase steadily beyond n = 20,000 because of the influence of rounding errors. With compensated summation the errors en are much less affected by rounding errors and do not grow in the range of n shown (for n = 108, en is about 10 times larger than it would be in exact arithmetic). Plots of U-shaped curves showing total error against stepsize are common in numerical analysis textbooks (see, e.g., Forsythe, Malcolm, and Moler [395, 1977, p. 119] and Shampine [910, 1994, p. 259]), but the textbooks rarely point out that the “U” can be flattened out by compensated summation.
96
Figure 4.2. Errors |y(1) compensated summation.
SUMMATION
for Euler’s method with (“×”) and without (“o”)
The cost of applying compensated summation in an ordinary differential equation solver is almost negligible if the function f is at all expensive to evaluate. But, of course, the benefits it brings are noticeable only when a vast number of integration steps are taken. Very long-term integrations are undertaken in celestial mechanics, where roundoff can affect the ability to track planetary orbits. Researchers in astronomy use compensated summation, and other techniques, to combat roundoff. An example application is a 3 million year integration of the planets in the solar system by Quinn, Tremaine, and Duncan [854, 1991]; it used a linear multistep method of order 13 with a constant stepsize of 0.75 days and took 65 days of machine time on a Silicon Graphics 4D-25 workstation. See also Quinn and Tremaine [853, 1991] and Quinlan [851, 1994]. Finally, we describe an even more ingenious algorithm called doubly compensated summation, derived by Priest [844, 1992] from a related algorithm of Kahan. It is compensated summation with 2 extra applications of the correction process11 and it requires 10 instead of 4 additions per step. The algorithm is tantamount to simulating double precision arithmetic with single precision arithmetic; it requires that the summands first be sorted into 11 The algorithm should perhaps be called triply compensated summation, but we adopt Priest’s terminology.
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decreasing order, which removes the need for certain logical tests that would otherwise be necessary. Algorithm 4.3 (doubly compensated summation). Given floating point numbers x 1,. . . , xn , this algorithm forms the sum by doubly compensated summation. All expressions should be evaluated in the order specified by the parentheses. Sort the xi so that |x i| > |x 2| > . . . > |x n |. s 1 = x1; c1 = 0 for i = 2:n yk = ck-1 + xk u k = xk - (yk - ck- 1 ) tk = yk + sk- 1 υ k = yk - (tk - sk-l ) zk = u k + υ k sk = tk + zk ck = zk - (sk - tk) end Priest [844, 1992, §4.1] analyses this algorithm for t-digit base β arithmetic that satisfies certain reasonable assumptions-ones which are all satisfied by IEEE arithmetic. He shows that if n < β t -3 then the computed sum satisfies that is, the computed sum is accurate virtually to full precision.
4.4. Other Summation Methods We mention briefly two further classes of summation algorithms. The first builds the sum in a series of accumulators, which are themselves added to give the sum. As originally described by Wolfe [1107, 1964] each accumulator holds a partial sum lying in a different interval. Each term xi is added to the lowestlevel accumulator; if that accumulator overflows it is added to the next-highest one and then reset to zero, and this cascade continues until no overflow occurs. Modifications of Wolfe’s algorithm are presented by Malcolm [723, 1971] and Ross [880, 1965]. Malcolm [723, 1971] gives a detailed error analysis to show that his method achieves a relative error of order u. A drawback of the algorithm is that it is strongly machine dependent. An interesting and crucial feature of Malcolm’s algorithm is that on the final step the accumulators are summed by recursive summation in order of decreasing absolute value, which in this particular situation precludes severe loss of significant digits and guarantees a small relative error.
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Another class of algorithms, referred to as “distillation algorithms” by Kahan [633, 1987], work as follows: given xi = fl( x i ), i = 1:n, they iteratively construct floating point numbers such that terminating when approximates with relative error at most u. Kahan states that these algorithms appear to have average run times of order at least n log n. See Bohlender [130, 1977], Kahan [633, 1987], Leuprecht and Oberaigner [700, 1982], Pichat [830, 1972], and Priest [844, 1992, pp. 66-69) for further details and references.
4.5. Statistical Estimates of Accuracy The rounding error bounds presented above can be very pessimistic, because they account for the worst-case propagation of errors. An alternative way to compare summation methods is through statistical estimates of the error, which may be more representative of the average case. A statistical analysis of three summation methods has been given by Robertazzi and Schwartz [874, 1988] for the case of nonnegative xi . They assume that the relative errors in floating point addition are statistically independent and have zero mean and finite variance Two distributions of nonnegative xi are considered: the uniform distribution on [0,2µ ]. and the exponential distribution with mean µ. Making various simplifying assumptions Robertazzi and Schwartz estimate the mean square error (that is, the variance of the absolute error) of the computed sums from recursive summation with random, increasing and decreasing orderings, and from insertion summation and pair-wise summation (with the increasing ordering). Their results for the summation of n numbers are given in Table -1.1. The results show that for recursive summation the ordering affects only the constant in the mean square error, with the increasing ordering having the smallest constant and the decreasing ordering the largest; since the xi are nonnegative, this is precisely the ranking given by the rounding error bound (4.3). The insertion and pairwise summation methods have mean square errors proportional to n 2 rather than n 3 for recursive summation, and the insertion method has a smaller constant than pairwise summation. This is also consistent with the rounding error analysis, in which for nonnegative xi the insertion met hod satisfies an error bound no larger than pairwise summation and the latter method has an error bound with a smaller constant than for recursive summation (log2 n versus n).
4.6. Choice of Method There is a wide variety of summation methods to choose from. For each met hod the error can vary greatly with the data, within the freedom afforded
4.6 C HOICE
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M ETHOD
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Table 4.1. Mean square errors for nonnegative xi . Distrib. Unif(0,2µ) EXP(µ)
Increasing 0.20µ 2 n 3 σ2 0.13µ 2 n 3 σ2
Random 0.33µ 2 n 3 σ2 0.33µ 2 n 3 σ2
Decreasing 0.53µ 2 n 3 σ2 0.63µ 2 n 3 σ2
Insertion 2.6µ 2 n 2 σ2 2.6µ 2 n 2 σ2
Pairwise 2.7µ 2 n 2 σ2 4.0µ 2 n 2 σ2
by the error bounds; numerical experiments show that, given any two of the methods, data can be found for which either method is more accurate than the other [553, 1993]. However, some specific advice on the choice of met hod can be given. 1. If high accuracy is important, consider implementing recursive summation in higher precision; if feasible this may be less expensive (and more accurate) than using one of the alternative methods at the working precision. What can be said about the accuracy of the sum computed at higher precision? If Sn = xi is computed by recursive summation at double precision (unit roundoff u 2) and then rounded to single precision, an error bound of the form holds. Hence a relative error of order u is guaranteed if nu Priest [844, 1992, pp. 62-63] shows that if the xi are sorted in decreasing order of magnitude before being summed in double precision, then holds provided only that n < β t -3 for t-digit base β arithmetic satisfying certain reasonable assumptions. Therefore the decreasing ordering may be worth arranging if there is a lot of cancellation in the sum. An alternative to extra precision computation is doubly compensated summation, which is the only other method described here that guarantees a small relative error in the computed sum. 2. For most of the methods the errors are, in the worst case, proportional to n. If n is very large, pairwise summation (error constant log2 n) and compensated summation (error constant of order 1) are attractive. 3. If the xi all have the same sign then all the methods yield a relative error of at most nu and compensated summation guarantees perfect relative accuracy (as long as nu < 1). For recursive summation of one-signed data, the increasing ordering has the smallest error bound (4.3) and the insertion method minimizes this error bound over all instances of Algorithm 4.1. 4. For sums with heavy cancellation recursive summation with the decreasing ordering is attractive, although it cannot be guaranteed to achieve the best accuracy.
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Considerations of computational cost and the way in which the data are generated may rule out some of the met hods. Recursive summation in the natural order, pairwise summation, and compensated summation can be implemented in O(n) operations for general xi , but the other methods are more expensive since they require searching or sorting. Furthermore, in an application such as the numerical solution of ordinary differential equations, where xk is not known until xi has been formed. sorting and searching may be impossible.
4.7. Notes and References This chapter is based on Higham [553, 1993]. Analysis of Algorithm 4.1 and compensated summation can also be found in Espelid [356, 1978]. The earliest error analysis of summation is that of Wilkinson for recursive summation in [1084, 1960], [1088, 1963]. Pairwise summation was first discussed by McCracken and Dorn [743, 1964, pp. 61-63], Babuska [35, 1969], and Linz [707, 1970]. Caprani [185, 1971] shows how to implement the method on a serial machine using temporary storage of size [log2 n] + 1 (without overwriting the xi ). The use of compensated summation with a Runge-Kutta formula is described by Vitasek [1055, 1969]. See also Butcher [170, 1987, pp. 118-120] and the experiments of Linnainrnaa [703, 1974]. Davis and Rabinowitz [267, 1984, §4.2.1] discuss pairwise summation and compensated summation in the context of quadrature.
Problems 4.1. Define and evaluate a condition number C(x) for the summation Sn (x) = When does the condition number take the value 1? 4.2. (Wilkinson [1088, 1963, p. 19]) Show that the bounds (4.3) and (4.4) are nearly attainable for recursive summation. (Hint: assume u = 2- t , set n = 2r (r |x2| > |x3| then compensated summation computes the sum x 1 + x2 + x3 with a relative error of order u (under reasonable assumptions on the arithmetic, such as the presence of a guard digit). He also gives the example x 1 = 2t
+ 1
,
x2=2 t+ 1 - 2 , x3= x4 = x5 = x 6 =-(2 t -1),
for which the exact sum is 2 but compensated summation computes 0 in IEEE single precision arithmetic (t=24). What is the smallest n for which compensated summation applied to x1,. . . , xn ordered by decreasing absolute value can produce a computed sum with large relative error?
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Chapter 5
Polynomials
The polynomial (z - 1)(z - 2) . . . (z - 20) is not a ‘difficult’ polynomial per se . . . The ‘difficulty’ with the polynomial Π(z - i) is that of evaluating the explicit polynomial accurately. If one already knows the roots, then the polynomial can be evaluated without any loss of accuracy. -J. H. WILKINSON, The Perfidious Polynomial (1984)
I first used backward error analysis in connection with simple programs for computing zeros of polynomials soon after the PILOT ACE came into use. -J. H. WILKINSON, The State of the Art in Error Analysis (1985)
The Fundamental Theorem of Algebra asserts that every polynomial equation over the complex field has a root. It is almost beneath the dignity of such a majestic theorem to mention that in fact it has precisely n roots. -J. H. WILKINSON, The Perfidious Polynomial (1984)
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Two common tasks associated with polynomials are evaluation and interpolation: given the polynomial find its values at certain arguments, and given the values at certain arguments find the polynomial. We consider Horner’s rule for evaluation and the Newton divided difference polynomial for interpolation. A third task not considered here is finding the zeros of a polynomial. Much research was devoted to polynomial zero finding up until the late 1960s; indeed, Wilkinson devotes a quarter of Rounding Errors in Algebraic Processes [1088, 19 6 3 ] to the topic. Since the development of the QR algorithm for finding matrix eigenvalues there has been less demand for polynomial zero finding, since the problem either arises as, or can be converted to (see §26.6 and [346, 1995], [1007, 1994]), the matrix eigenvalue problem.
5.1. Horner’s Method The standard met hod for evaluating a polynomial p(x) = a0 + a 1 x + . . . + anxn
(5.1)
is Horner’s method (also known as Horner’s rule and nested multiplication), which consists of the following recurrence: qn(x) = an for i = n - 1: -1:0 qi(x) = xqi+ 1 (x) + ai end p(x) = q0(x) The cost is 2n flops, which is n less than the more obvious method of evaluation that explicitly forms powers of x (see Problem 5.2). To analyse the rounding errors in Horner’s met hod it is convenient to use the relative error counter notation (see (3.9)). We have
It is easy to either guess or prove by induction that
(5.2)
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5.1 H ORNER'S M ETHOD
where we have used Lemma 3.1, and where |φ k | < ku/(1 - ku) =: γk . This result shows that Horner’s method has a small backward error: the computed is the exact value at x of a polynomial obtained by making relative perturbations of size at most γ2 n to the coefficients of p( x). A forward error bound is easily obtained: from (5.2) we have (5.3) where
The relative error is bounded according to
Clearly, the factor ψ ( p , x ) can be arbitrarily large. However, ψ(p,x) = 1 if ai > 0 for all i and x > 0, or if (-l)i ai > 0 for all i and x < 0. In a practical computation we may wish to compute an error bound along with The bound (5.3) is entirely adequate for theoretical purposes and can itself be computed by Horner’s method. However, it lacks sharpness for two reasons. First, the bound is the result of replacing each γ k by γ2 n . Second, and more importantly, it is an a priori bound and so takes no account of the actual rounding errors that occur. We can derive a sharper, a posteriori bound by a running error analysis. For the ith step of Horner’s method we can write (5.4) where we have used both (2.4) and (2.5). Defining
=: qi + fi , we have
or Hence Since fn = 0, we have |fi | < uπi , where
We can slightly reduce the cost of evaluating the majorizing sequence πi working with which satisfies the recurrence
We can now furnish Horner’s method with a running error bound.
by
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Algorithm 5.1. This algorithm evaluates y = fl(p(x)) by Horner’s method, It also evaluates a quantity µ such that |y - p(x)| < µ. where p(x) = y = an µ = |y|/2 for i = n- 1 : - 1 : 0 y = xy + ai µ = |x|µ + |y | end µ = µ( 2µ - |y|) Cost: 4n flops. It is worth commenting on the case where one or more of the ai and x is complex. The analysis leading to Algorithm 5.1 is still valid for complex data, but we need to remember that the error bounds for fl ( x op y) are not the same as for real arithmetic. In view of Lemma 3.5. it suffices to replace the last line of the algorithm by µ = An increase in speed of the algorithm, with only a slight worsening of the bound, can be obtained by replacing |y| = (( Rey )2 + (Imy)2)½ by |Rey| + |Imy| (and, of course, |x| should be evaluated once and for all before entering the loop). One use of Algorithm 5.1 is to provide a stopping criterion for a polynomial zero-finder: if |fl(p ( x))| is of the same order as the error bound µ, then further iteration serves no purpose, for as far as we can tell, x could be an exact zero. As a numerical example, for the expanded form of p (x) = (x + 1)32 we found in MATLAB that
and for p(x) the Chebyshev polynomial of degree 32,
In these two cases. the running error bound is, respectively. 62 and 31 times smaller than the a priori one. In another experiment we evaluated the expanded form of p(x) = (x - 2)3 is simulated single precision in MATLAB (u 6 × 10-8) for 200 equally spaced points near x = 2. The polynomial values, the error, and the a priori and running error bounds are all plotted in Figure 5.1. The running error bound is about seven times smaller than the a priori one.
5.2. Evaluating Derivatives Suppose now that we wish to evaluate derivatives of p. We could simply differentiate (5.1) as many times as necessary and apply Horner’s method to
5.2 E VALUATING D ERIVATIVES
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Figure 5.1. Computed polynomial values (top) and running and a priori bounds (bottom) for Horner’s method. each expression, but there is a more efficient way. Observe that if we define q (x) = q 1 + q 2 x + . . . + qnxn -1 r = q0, where the qi = qi (a) are generated by Horner’s method for p(a), then p(x) = (x - a)q(x) + r. In other words, Horner’s method carries out the process of synthetic division. Clearly, p'(a) = q(a). If we repeat synthetic division recursively on q(x), we will be evaluating the coefficients in the Taylor expansion
and after a final scaling by factorials, we will obtain the derivatives of p at a . The resulting algorithm is quite short. Algorithm 5.2. This algorithm evaluates the polynomial and its first k derivatives at a, returning yi = p( i)(a), i = 0:k. y0 = an y(1:k) = 0
P OLYNOMIALS
108 for j = n- 1 : - 1 : 0 for i = min(k, n - j): -1:1 yi = ayi + yi- 1 end y 0 = ay0 + aj end m = l for j = 2:k m = m*j yj = m*yj end
cost: nk + 2(k + n) - k2/2 flops. How is the error bounded for the derivatives in Algorithm 5.2? To answer this quest ion with the minimum of algebra, we express the algorithm in matrix notation. Horner’s method for evaluating p(a) is equivalent to solution of the bidiagonal system
By considering (5.4). we see that
Hence (5.5) The recurrence for r0 = p'(a) can be expressed as Unr = q(1:n), where r = r(0:n - 1), so
Hence
This gives, using (5.5), (5.6)
5.3 T HE N EWTON FORM
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P OLYNOMIAL I NTERPOLATION
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Now
By looking at the form of r and q, we find from (5.6) that
(5.7) This is essentially the same form of bound as for p(a) in (5.3). Analogous bounds hold for all derivatives.
5.3. The Newton Form and Polynomial Interpolation An alternative to the monomial representation of a polynomial is the Newton form (5.8) which is commonly used for polynomial interpolation. The interpolation problem is to choose p so that p(a i ) = fi, i = 0:n, and the numbers ci are known
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as divided differences. Assuming that the points a j are distinct, the divided differences may be computed from a standard recurrence: c(0)(0: n) = f( 0 :n ) for k = 0:n - 1 f o r j = n:-1:k +1 end end c = c( n ) Cost: 3n 2/2 flops. Two questions are of interest: how accurate are the computed and what is the effect of rounding errors on the polynomial values obtained by evaluating the Newton form? To answer the first question we express the recurrence in matrix-vector form: c(0)= f , where Lk =
c( k + l ) =L k c( k ),
k=0:n-1,
is lower bidiagonal, with
Dk = diag(ones(1:k + 1), ak+1
- a 0 ,a k+2 - a1, . . . ,an - an -k -1),
The analysis that follows is based on the model (2.4), and so is valid only for machines with a guard digit. With the no-guard-digit model (2.6) the bounds become weaker and more complicated, because of the importance of terms f l(aj - aj-k-1) in the analysis. It is straightforward to show that (5.9) where Gk = diag(ones(1:k + 1), η k , k+2,. . . ,η k , n+1), where each η ij is of the form η ij = (1 + δ1)(1 + δ2)(1 + δ3), |δi | < u. Hence (5.10) From Lemma 3.7.
(5.11)
5.3 T HE N EWTON F ORM
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111
To interpret the bound, note first that merely rounding the data (f i f i (1 + δi ), |δ i | < u) can cause an error ∆c as large as eround = u|L||f|, where L = Ln-1. . . L0, so errors of at least this size are inevitable. Since |L n-1 | . . . |L0 | > |Ln- 1 . . .L 0 | = |L|, the error in the computed divided differences can be larger than eround only if there is much subtractive cancellation in the product L = Ln-1 . . . L0. If a 0 < a 1 < . . . < an , then each Li is positive on the diagonal and nonpositive on the first subdiagonal; therefore |L n-1|. . . |L0| = |Ln-1. . . L0| = |L|, and we have the very satisfactory bound < ( ( 1-3u ) -n - 1)|L||f|. This same bound holds if the ai are arranged in decreasing order. To examine how well the computed Newton form reproduces the f i we “unwind” the analysis above. From (5.9) we have
By invoking Lemma 3.7 again, we obtain (5.12) If a 0 < a1 < ... < an then > 0 for all i, and we obtain the very satisfactory bound Again, the same bound holds for points arranged in decreasing order. In practice it is found that even when the computed divided differences are very inaccurate, the computed interpolating polynomial may still reproduce the original data well. The bounds (5.11) and (5.12) provide insight into this observed behaviour by showing that and can be large only when there is much cancellation in the products Ln-l . . . L0 f and respectively. The analysis has shown that the ordering a 0 < a 1 < . . . < an yields “optimal” error bounds for the divided differences and the residual, and so may be a good choice of ordering of interpolation points. However, if the aim is to minimize |p(x) - fl(p(x))| for a given x aj, then other orderings need to be considered. An ordering with some theoretical support is the Leja ordering, which is defined by the equations [863, 1990] (5.13a) (5.13b) For a given set of n + 1 points ai , the Leja ordering can be computed in n 2 flops (see Problem 5.4). We give a numerical example to illustrate the analysis. Let n = 16 and let a 0 < . . . < an be equally spaced points on [-1,1] . Working in simulated
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single precision with u = 2-24 6 × 10 -8 , we computed divided differences for two different vectors f. Error statistics were computed by regarding the solutions computed in double precision as exact. We define the ratios
(1) For f i from the normal N(0,1) distribution the divided differences range in magnitude from 1 to 105. and their relative errors range from 0 (the first divided difference, f 0 , is always exact) to 3 × 10-7. The ratio p1 = 16.3, so (5.11) provides a reasonably sharp bound for the error in . The relative errors when f is reconstructed from the computed divided differences range between 0 and 3 × 10-1 (it makes little difference whet her the reconstruction is done in single or double precision). Again, this is predicted by the analysis, in this case by (5.12), because p 2 = 2 × 107. For the Leja ordering, the divided differences are computed with about the same accuracy, but f is reconstructed much more accurately, with maximum relative error 7 × 10-6 ( p 1 = 1 × 103, p2 = 8 × 104). (2) For f i = exp(a i), the situation is reversed: we obtain inaccurate divided differences but an accurate reconstruction of f. The divided differences range in magnitude from 10-4 to 10-1, and their relative errors are as large as 1, but the relative errors in the reconstructed f are all less than 10-7. Again, the error bounds predict this behaviour: p 1 = 6 × 108, p2 = 1.02. The Leja ordering performs similarly. The natural way to evaluate the polynomial (5.8) for a given x is by a generalization of Horner’s met hod: q n(x ) = cn for i = n - 1:-1:0 q i ( x) = ( x- a i )q i + 1 ( x) + ci end p (x) = q 0 (x) A straightforward analysis shows that (cf. (5.2))
Hence the computed is the exact value corresponding to a polynomial with slightly perturbed divided differences. The corresponding forward error bound is
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AND
R EFERENCES
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5.4. Notes and References Backward and forward error analysis for Horner’s rule was given by Wilkinson [1088, 1963, pp. 36-37, 49-50]; our results are simply Wilkinson’s presented in a different notation. The analysis has been redone by many other authors, sometimes without reference to Wilkinson’s results. Another early reference, which gives a forward error bound only, is McCracken and Dorn [743, 1964, §3.5], For more on running error analysis see §3.3. Müller [782, 1983] gives a first-order error analysis for the evaluation of the divided difference form of a polynomial. Olver [809, 1986] derives a posteriori error bounds for the Horner scheme with derivatives (Algorithm 5.2), phrasing them in terms of his relative precision notation. Stewart [939, 1971] analyses synthetic division, using a matrix-oriented approach similar to that in §5.2. The relative merits of the monomial and Chebyshev representations of a polynomial are investigated, with respect to accuracy of evaluation, by Newbery [793, 1974] and Schonfelder and Razaz [901, 1980]. Clenshaw [212, 1955] showed how Horner’s method could be extended to evaluate a polynomial expressed in the Chebyshev form p(x) = where Ti is the Chebyshev polynomial of degree i. Error analysis of Clenshaw’s method, and variations of it, are given by Gentleman [433, 1969], Newbery [792, 1973], and Oliver [805, 1977], [806, 1979]. Clenshaw’s scheme can be generalized to expansions in terms of arbitrary orthogonal polynomials; see Smith [927, 1965] and Algorithm 21.8. Running error bounds for Horner’s method were included in algorithms of Kahan and Farkas [637, 196 3], [638, 196 3] without explanation. Adams [5, 1967] derives the bounds and extends them to evaluation of a real polynomial at a complex argument. Algorithm 5.1 is given in [5, 1967], and also in the classic paper by Peters and Wilkinson [827, 1971], which describes many aspects of the solution of polynomial equations. Wilkinson’s paper “The Perfidious Polynomial” [1103, 1984] (for which he was awarded the Chauvenet Prize) is highly recommended as a beautifully written introduction to backward error analysis in general and error analysis for polynomials in particular. There seems to be little work on choosing the ordering of interpolation points to minimize the effect of rounding errors on the construction or evaluation of the interpolating polynomial. Werner [1075, 1984] examines experimentally the effect of different orderings on the computed value of an interpolating polynomial at a single point, for several forms of interpolating polynomial. The Leja ordering, which was proposed by Leja in a 1957 paper, is analysed in detail by Reichel [863, 1990 ]. He shows that an appropriately defined condition number for the Newton form of interpolating polynomial grows at a slower than exponential rate in the degree n for Leja points, which are points
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POLYNOMIALS
taken from a given compact set that satisfy the condition (5.13). For more on the numerical benefits of the Leja ordering see §21.3.3. If a polynomial is to be evaluated many times at different, arguments it may be worthwhile to expend some effort transforming it to a form that can be evaluated more cheaply than by a straightforward application of Horner’s rule. For example, the quartic p(x) = a 4 x4 + a 3 x3 + a 2 x2 + a 1 x + a 0 , can be rewritten as [668,
1981,
a4
0,
p. 471]
p(x) = (( y + x + a2 ) y + a 3 )a4 ,
y = (x + a 0 )x + a 1 ,
where the coefficients ai are given by.
Once the ai have been computed, p(x ) can he evaluated in three multiplications and five additions, as compared with the four multiplications and four additions required by Horner’s rule. If a multiplication takes longer than an addition, the transformed polynomial should be cheaper to evaluate. For polynomials of degree n > 4 there exist evaluation schemes that require strictly less than the 2n total additions and multiplications required by Horner’s rule; see Knuth [665, 196 2], [668, 198 1, pp. 471-475] and Fike [373, 196 7] One application in which such schemes have been used is in evaluating polynomial approximations in an elementary function library [412, 1991]. Little seems to be known about the numerical stability of fast polynomial evaluation schemes; see Problem 5.6.
Problems 5.1. Give an alternative derivation of Algorithm 5.2 by differentiating the Horner recurrence and rescaling the iterates. 5.2. Give an error analysis for the following “beginner’s” algorithm for evaluating p(x) = a 0 + a1 x + . . . + anxn : q (x) = a 0; y = 1 for i = 1:n y = xy q(x) = q(x) + ai y end p(x) = q(x )
PROBLEMS
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5.3. Let p(x) = a 0 + a 1 x + . . . + anxn and n = 2 m. Then p(x) = (a 0 + a 2 x2 + . . . + a 2 m x2 m ) + (a 1 x + a 3 x3 + . . . + a 2 m - 1 x2 m - 1 ) = a0 + a 2 y + . . . + a2 mym + x(a 1 + a3 y + . . . + a 2 m - 1 y m- 1 ), where y = x2. Obtain an error bound for fl(p(x)) when p is evaluated using this splitting (using Horner’s rule on each part). 5.4. Write down an algorithm for computing the Leja ordering (5.13) in n 2 flops. 5.5. If the polynomial p(x) = has roots x1,. . . , xn , it can be evaluated from the root product form p(x) = Give an error analysis for this evaluation. 5.6. (RESEARCH PROBLEM) Investigate the numerical stability of fast polynomial evaluation schemes (see the Notes and References) by both rounding error analysis and numerical experiments. For a brief empirical study see Miller [757, 1975, §10].
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Chapter 6
Norms
While it is true that all norms are equivalent theoretically, on/y a homely one like the -norm is truly useful numerically. -J. H. WILKINSON 12, Lecture at Stanford University (1984)
Matrix norms are defined in many different ways in the older literature, but the favorite was the Euclidean norm of the matrix considered as a vector in n 2-space. Wedderburn (1934) calls this the absolute value of the matrix and traces the idea back to Peano in 1887. -ALSTON S. HOUSEHOLDER, The Theory of Matrices in Numerical Analysis (1964)
12
Quoted in Fox [403,
19 8 7 ].
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ORMS
Norms are an indispensable tool in numerical linear algebra. Their ability to compress the mn numbers in an m × n matrix into a single scalar measure of size enables perturbation results and rounding error analyses to be expressed in a concise and easily interpreted form. In problems that are badly scaled, or contain a structure such as sparsity, it is often better to measure matrices and vectors componentwise. But norms remain a valuable instrument for the error analyst, and in this chapter we describe some of their most useful and interesting properties.
6.1. Vector Norms A vector norm is a function ||·|| :
IR satisfying the following conditions:
1. ||x|| > 0 with equality iff x = 0. 2. ||ax|| = |a| ||x|| for all 3. ||x+y|| < ||x|| + ||y|| for all x, y
(the triangle inequality).
The three most useful norms in error analysis and in numerical computation are “Manhattan“ or “taxi cab” norm, (x*x)½.
Euclidean length,
These are all special cases of the Holder y-norm:
The 2-norm has two properties that make it particularly useful for theoretical purposes. First, it is invariant under unitary transformations, for if Q*Q = I, then = x*Q*Qx = x*x = Second, the 2-norm is differentiable for all x, with gradient vector ||x||2 = x/||x||2. A fundamental inequality for vectors is the Hölder inequality (see, for example, [502, 1967, App. 1]) |x*y| < ||x||p ||y||q ,
(6.1)
119
6.1 V ECTOR N ORMS This is an equality when p, dependent and xi yi lies on is also possible when p = with p = q = 2 is called
q > 1 if the vectors (|x i |p ) and (|y i |q ) are linearly the same ray in the complex plane for all i; equality 1 and p = , as is easily verified. The special case the Cauchy-Schwarz inequality: |x*y| < ||x||2 ||y||2.
For an arbitrary vector norm ||·|| the dual norm is defined by (6.2) It follows from the Hölder inequality that the dual of the p -norm is the q-norm, where p- 1 +q -1 = 1. The definition of dual norm yields, trivially, the general Hölder inequality |x*y| < ||x|| ||y||D . For a proof of the reassuring result that the dual of the dual norm is the original norm (the “duality theorem”) see Horn and Johnson [580, 1985, Thm. 5.5.14]. In some analyses we need the vector z dual to y, which is defined by the property (6.3) z*y = ||z||D ||y || = 1. That such a vector z exists is a consequence of the duality theorem (see [580, 1985, Cor. 5.5.15]). How much two p-norms of a vector can differ is shown by the attainable inequalities [422, 1983, pp. 27-28], [459, 1983, Lem. 1.1] (6.4) The p-norms have the properties that ||x|| depends only on the absolute value of x, and the norm is an increasing function of the absolute values of the entries of x. These properties are important enough to warrant a definition. Definition 6.1. A norm on 1. monotone if |x| < |y|
is ||x|| < ||y|| for all x, y
and
2. absolute if || |x| || = ||x|| for all x The following nonobvious theorem shows that these two properties are equivalent. Theorem 6.2 (Bauer, Stoer, and Witzgall). A norm on and only if it is absolute. Proof. See Horn and Johnson [580, Sun [954, 1990, Thm. 2.1.3].
19 8 5 ,
is monotone if
Thm. 5.5.10], or Stewart and
120
NORMS
6.2. Matrix Norms A matrix norm is a function ||·|| : IR satisfying obvious analogues of the three vector norm properties. The simplest example is the Frobenius norm.
(which is sometimes called the Euclidean norm and denoted ||·||E ). A very important class of matrix norms are those subordinate to vector norms. Given a vector norm on the corresponding subordinate matrix norm on is defined by (6.5) or, equivalently,
(Strictly speaking, this definition uses two different norms: one on in the numerator of (6.5) and one on in the denominator. Thus the norm used in the definition is assumed to form a family defined on for any s.) For the l-, 2-, and -vector norms it can be shown that
where the spectral radius
and where σm a x (A) denotes the largest singular value of A. To remember the formulae for the 1- and -norms, note that 1 is a vertical symbol (for columns) and is a horizontal symbol (for rows). A norm is consistent if it satisfies ||AB|| < ||A|| ||B|| whenever the product AB is defined. The Frobenius norm and all subordinate norms are consistent. An example of a norm that is not consistent is the “max norm” ||A|| = maxi , j |a ij |. The best bound that holds for all A and B is ||AB|| < n||A|| ||B||, with equality when aij 1 and bij 1.
121
6.2 MATRIX NORMS
Table 6.1. Constants apq such that ||x||p < apq||x||q, x
A norm for which ||UAV|| = ||A || for all unitary U and V is called a unitarily invariant norm. These norms have an interesting theory, which we will not explore here (see [581, 1991, §3.5] or [954, 1990, §2.3]). Only two unitarily invariant norms will be needed for our analysis: the 2-norm and the Frobenius norm. That these two norms are unitarily invariant follows easily from the formulae above. For any unitarily invariant norm, the useful property holds that ||A*|| = ||A||. The 2-norm satisfies the additional relation ||A*A||2 = ||A|| . The unitary invariance of the 2- and Frobenius norms has implications for error analysis, for it means that multiplication by unitary matrices does not magnify errors. For example, if A is contaminated by errors E and Q is unitary, then Q(A+E)Q* = QAQ*+F, and ||F||2 = ||QEQ*||2 = ||E||2. In contrast, if we do a general, nonsingular similarity transformation X(A+E)X -1 = XAX- 1 +G, then ||G||2 = ||XEX-1||2 < κ 2 (X)||E||2, where κ(X) = ||X|| ||X -1|| is the condition number of X. The condition number satisfies κ(X) > 1 (κF (X) > and can be arbitrarily large. In perturbation theory and error analysis it is often necessary to switch between norms. Therefore inequalities that bound one norm in terms of another are required. It is well known that on a finite-dimensional space any two norms differ by at most a constant that depends only on the dimension (so-called norm equivalence). Tables 6.1 and 6.2 give attainable inequalities for the vector and matrix norms of most interest. The definition of subordinate matrix norm can be generalized by permitting different norms on the input and output space: (6.6)
122
Table 6.2. Constants apq such that ||A||p < apg||A||q, A max i,j |aij| and ||A||s :=
NORMS
Here, ||A||M :=
Note that, in general, the submultiplicative property ||AB||a, β < ||A||a,β||B||a, β does not hold. but we do have (6.7) for any third vector norm ||·||γ. The choice a = 1 and β = produces the max norm, mentioned above, ||A||1, = maxi , j |a ij |. At least two important results about matrix norms hold for this mixed subordinate norm. The first is a simple formula for the matrix condition number of a nonsingular A defined by
Note that this definition uses the ||·||a,β norm on the data space and the ||·||β,a norm on the solution space, as is natural. We need the following lemma. Lemma 6.3. Given vector norms ||·||a and ||·||β and vectors x, y such that ||x||a = ||y||β = 1, there exists a matrix B with ||B||a,β = 1 such that Bx = y. Proof. Recall that the dual of the a-norm is defined max||w ||a= 1 |z*w|. Let z be a vector dual to x, so that z*x = and hence Let B = yz*. Then Bx = y and
as required.
by
123
6.2 M A T R I X N O R M S Theorem 6.4. For nonsingular A κa , β (A) satisfies
the matrix condition number
κa ,β (A) = ||A||a , β ||A -1 || β , a .
(6.8)
Proof. In view of the expansion (A+ ∆A) - 1 - A- 1 = - A- 1 ∆AA - 1 + O(||∆A|| 2 ), the result is proved if we can show that (6.9) That (6.9) holds with the equality replaced by “ ||A||p 2. The result (6.15) implies, for example, that for all p > 1 (6.16) (6.17)
125
6.3 THE MATRIX p- N O R M
Figure 6.1. Plots of p versus ||A||p, for 1 < p < 15. Fourth plot shows 1/ p versus log ||A||p for the matrices in the first three plots. Upper bounds for ||A||p that do not involve m or n can be obtained from the interesting property that log ||A||p is a convex function of 1/p for p > 1 (see Figure 6.1), which is a consequence of the Riesz-Thorin theorem [503, 1952, pp. 214, 219], [450, 1991]. The convexity implies that if f(a) = ||A||1 / a , then for 0 < a,β < 1, logf(θ a + (1 - θ)β) < θ logf(a) + ( 1 - θ1) logf(β),
0 < θ < l.
Writing p 1 = 1/a and p 2 = 1/β, this inequality can be expressed as (6.18)
Two interesting special cases are (6.19) and (6.20) Note that (6.19) includes the well-known inequality
NORMS
126 Two further results that are familiar for p = 1, 2,
are (6.21)
(see, for example, [580,
1985,
Thm. 5.6.36]) and
The bounds (6.16) and (6.17) imply that given the ability to compute ||A||1, ||A||2 and ||A|| we can estimate ||A||p correct to within a factor n 1 / 4 . These a priori estimates are at their best when p is close to 1, 2, or but in general they will not provide even one correct significant digit. The bound in (6.18) can be much smaller than the other upper bounds given above, but how tight it is depends on how nearly log ||A||p is linear in p . Numerical methods are needed to obtain better estimates: these are developed in chapter 14.
6.4. Notes and References The matrix condition number appears to have been first introduced explicitly by Turing [1027, 1948], who defined, for example, the N-condition number of A IRn×x as n -1 N(A)N(A -1), where N(·) is Turing’s notation for the Frobenius norm. Todd [1003, 1968] gives a short survey of the matrix condition number with many references. Theorem 6.2 was originally proved by Bauer, Stoer, and Witzgall, in a paper that contains many interesting results on monotonic norms [84, 1961]. Tables of constants in inequalities between different norms have been given by various authors: see, for example, Stone [957, 1962] and Zielke [1129, 1988]. Our development of the mixed subordinate norm ||·|| a ,β is based on that of D. J. Higham [526, 1995]. Theorem 6.5 is proved by Kahan [626, 1966, pp. 775 776], who attributes it to Gastinel but gives no reference. For the 2-norm, this result goes back to a paper by Eckart and Young [334, 1936]. Theorem 6.5 is an instance of a relationship that holds for many problems: the condition number is the reciprocal of the distance to the nearest singular problem (one with an infinite condition number). This relationship applies to matrix inversion, eigenvalue and eigenvector computation, polynomial zero-finding, and pole assignment in linear control systems. For an in-depth study see Demmel [281, 1987]. Direct proofs of inequality (6.19) can be found in Kato [646, 1976, p. 29] and Todd [1006, 1977, pp. 25-26]. The inequality does not seem to be well known. For historical comments on the development of norms in numerical analysis, see Householder [587, 1964, Chap. 2] and Stewart and Sun [954, 1990, Chap. 2].
127
PROBLEMS
Problems Problems worthy of attack prove their worth by hitting back. -PIET HEIN, Grooks (1966)
6.1. Prove the inequalities given in Tables 6.1 and 6.2. Show that each inequality in Table 6.2 (except the one for aS,2) is attainable for a matrix of the form A = xyT, where x, y {e, ej}, where e = [1, 1,. . . , 1]T. Show that equality in ||A||s < as,2 ||A||2 is attained for square real matrices A iff A is a scalar multiple of a Hadamard matrix (see §9.3 for the definition of a Hadamard matrix), and for square complex matrices if a rs = exp(2 πi(r - 1)(s - 1)/n) (this is a Vandermonde matrix based on the roots of unity). 6.2. Let x, y ||x|| ||y||D .
Show that, for any subordinate matrix norm, ||xy*|| =
6.3. Show that a subordinate matrix norm ||·|| on
satisfies
From ancient times until now the study of magic squares has flourished as a kind of cult, often with occult trappings, whose initiates range from such eminent mathematicians as Arthur Cayley and Oswald Veblen to laymen such as Benjamin Franklin. -MARTIN GARDNER, More Mathematical Puzzles and Diversions (1961)
6.4. Let Mn denote a magic square matrix, that is, an n × n matrix containing the integers from 1 to n 2 arranged in such a way that the row and column sums are all the same. Let µ n , denote the magic sum of Mn (thus, µn = n(n2 + 1)/2). Show that ||Mn||p = µ n for all 1 < p < (This result is a special case of an apparently little-known result of Stoer and Witzgall, which states that the norm of a doubly stochastic matrix is 1 for any norm subordinate to a permutation-invariant absolute vector norm [956, 1962].) 6.5. Show that ||ABC||F < ||A||2 ||B||F||C|| 2 for any A, B, and C such that the product is defined. (This result remains true when the Frobenius norm is replaced by any unitarily invariant norm [581, 1991, p. 211].)
NORMS
128 6.6. Show that for any nonsingular
6.7. Show that for any and any consistent matrix norm, p(A) < ||A||, where p is the spectral radius. 6.8. Show that for any and δ > 0 there is a consistent norm ||·|| (which depends on A and δ) such that ||A|| < p(A) + δ, where p is the spectral radius. Hence show that if p (A) < 1 then there is a consistent norm ||·|| such that ||A|| < 1. 6.9. Let Use the SVD to find expressions for ||A||2 and ||A||F in terms of the singular values of A. Hence obtain a bound of the form c1 ||A|| 2 < ||A|| F < c2 ||A|| 2, where c 1 and c2 are constants that depend on n. When is there equality in the upper bound? When is there equality in the lower bound? 6.10. Show that
Deduce that when ||F||2 = 1, the norm is 6.11. Let ||A||a, = maxi 6.12. (Tao [994,
(Rohn [879,
1995]
the golden ratio.
Prove that (a) ||A||1 , β = maxj ||A (i,:)|| What is ||A||1 , ? 1984])
||A (:,j)||β , and (b)
Show that if A is Hermitian positive definite then
shows that the problem of computing ||A|| ,1 is NP-hard.)
6.13. Prove that if H
IRn×n is a Hadamard matrix then ||H|| p = max{n1 /p ,n1 - 1 /p }.
(See §9.3 for the definition of a Hadamard matrix.) 6.14. Show that if A
IRm×n has at most µ nonzeros per row then (6.22)
129
PROBLEMS while if A has at most µ nonzeros per column then
(6.23) where p -1 + q-1 = 1. (These inequalities generalize (6.12) and (6.13).) 6.15. Show that if A
then for any p-norm (1 < p
0 is a right Perron vector of |A||A - 1 | (so that |A||A- 1 | = p(|A||A - 1 |)x). Proof. See Problem 7.9. For the Kahan example (7.14),
and, in fact, 3 for D = two-sided scaling is nearly optimal in this case.
so a symmetric
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FOR
LINEAR S YSTEMS
7.4. The Matrix Inverse We briefly discuss component wise perturbation theory for the matrix inverse. With X := A-1 and X + ∆X := (A + ∆X)-1, a componentwise condition number is
In general, the inequality- is strict. but there is equality when |A -1| = D1 A - 1D 2 for D i of the form diag(±1), [407, 1992, Thm. 1.10], [439, 198 2]. Another componentwise condition number is evaluated in Problem 7.10. We saw in Theorem 6.5 that the reciprocal of the normwise condition number for matrix inversion is the normwise relative distance to singularity. Is the same true for an appropriate componentwise condition number? The componentwise distance to singularity,
has been characterized by Rohn [877,
1989],
[878,
1990]
as
where the maximum is taken over all signature mat rices Si = diag(±1) and where This formula involves 4n eigenproblems and thus is computationally intractable (in fact it has been shown to be NP-hard by Poljak and Rohn [836, 1993]). Demmel [285, 1992] shows by complexity arguments that there can be no simple relationship between dE (A) and the quantity which is an upper bound for µ E (A). He also presents evidence for the conjecture that
for a constant γn . The lower bound always holds and Demmel identifies several classes of matrices for which the upper bound holds. This conjecture is both plausible and aesthetically pleasing because d|A| (A) is invariant under two-sided diagonal scalings of A and p(|A- 1 ||A|) is the minimum -norm condition number achievable by such scalings, as shown by Theorem 7.8.
7.5. Extensions The componentwise analyses can be extended in three main ways.
7.6 N UMERICAL S TABILITY
141
(1) We can use more general measures of size for the data and the solution. Higham and Higham [528, 1992] measure ∆A , ∆b, and ∆x by
where and the e ij , fi , and gi are tolerances. They show that the corresponding backward error is given by the explicit formula
where r = b - Ay, Dj = diag(e j1, . . . , ejn, fj), and p -1 + q -1 = 1; bounds for the corresponding condition number are also obtained. Theorem 7.3, and Theorem 7.4 with the -norm, correspond to p = and gi If we take p = and g = |x|, we are measuring the change in the solution in a componentwise relative sense, as and the condition number is [528, 1992]
This latter case has also been considered by Rohn [876, 1989] and Gohberg and Koltracht [455, 1993]. It is also possible to obtain individual bounds for by refraining from taking norms in the analysis: see Chandrasekaran and Ipsen [197, 1995] and Problem 7.1. (2) The backward error results and the perturbation theory can be extended to systems with multiple right-hand sides. For the general vp measure described in (1), the backward error can be computed by finding the minimum p-norm solutions to n underdetermined linear systems. For details, see Higham and Higham [528, 1992]. (3) Structure in A and b can be preserved in the analysis. For example, if A is symmetric or Toeplitz then its perturbation can be forced to be symmetric or Toeplitz too, while still using componentwise measures. References include Higham and Higham [527, 1992] and Gohberg and Koltracht [455, 1993] for linear structure, and Bartels and D. J. Higham [76, 1992] for Vandermonde structure. A symmetry-preserving normwise backward error is explored by Bunch, Demmel, and Van Loan [163, 1989], while Smoktunowicz [930, 1995] considers the componentwise case (see Problem 7.11). Symmetry-preserving normwise condition numbers are treated by D. J. Higham [526, 1995].
7.6. Numerical Stability The backward errors examined in this chapter lead to definitions of numerical stability of algorithms for solving linear systems. Precise and formal definitions of stability can be given, but there are so many possibilities, across
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PERTURBATION THEORY
FOR
LI N E A R S Y S T E M S
different problems, that to define and name each one tends to cloud the issues of interest. We therefore adopt an informal approach. A numerical method for solving a square, nonsingular linear system Ax = b is normwise backward stable if it produces a computed solution such that is of order the unit roundoff. How large we allow to be. while still declaring the method backward stable, depends on the context,. It is usually implicit in this definition that = O(u) for all A and b, and a method that yields = O(u) or a particular A and b is said to have performed in a normwise backward stable manner. The significance of normwise backward stability is that the computed solution solves a slightly perturbed problem, and if the data A and b contain uncertainties bounded only normwise with and similarly for b), then may be the exact solution to the problem we wanted to solve, for all we know. Componentwise backward stability is defined in a similar way: we now reto be of order u. This is a quire the componentwise backward error more stringent requirement than normwise backward stability. The rounding errors incurred by a met hod that is componentwise backward stable are in size and effect equivalent to the errors incurred in simply converting the data A and b to floating point numbers before the solution process begins. If a method is normwise backward stable then, by Theorem 7.2, the forward error is bounded by a multiple of κ(A )u. However, a met hod can produce a solution whose forward error is bounded in this way without the normwise backward error being of order u. Hence it is useful to define a method for which as normwise forward stable. By similar reasoning involving we say a method is componentwise forward stable if Table 7.1 summarizes the definitions and the relations between them. There are several examples in this book of linear-equation-solving algorithms that are forward stable but not backward stable: Cramer’s rule for n = 2 (§1.10.1). Gauss-Jordan elimination (§13.4), and the seminormal equations method for underdetermined systems (§20.3). Other definitions of numerical stability can be useful (for example, rowwise backward stability means that and they will be introduced when needed.
7.7. Practical Error Bounds Suppose we have a computed approximation to the solution of a linear system Ax = b, where What error bounds should we compute?
143
7.7 P RACTICAL E RROR B OUNDS
Table 7.1. Backward and forward stability. Componentwise backward stability
Componentwise forward stability
Normwise backward stability
Normwise forward stability
The backward error can be computed exactly, from the formulae
(7.23) at the cost of one or two matrix-vector products, for r = b and The only question is what to do if the denominator is so small as to cause overflow or division by zero in the expression for This could happen, for example, when E = |A| and f = |b| and, for some i , aijxj = 0 for all j, as is most likely in a sparse problem. LAPACK’s xyyRFS (“refine solution”) routines apply iterative refinement in fixed precision, in an attempt If the ith component of the denominator in (7.23) to satisfy is less than safe_min/u, where safe_min is the smallest number such that 1/safe_min does not overflow, then they add (n + 1) safe_min to the ith components of the numerator and denominator. A more sophisticated strategy is advocated for sparse problems by Arioli, Demmel, and Duff [24, 1989]. They suggest modifying the formula (7.23) by replacing |bi | in the denominator by when the ith denominator is very small. See [24, 1989] for details and justification of this strategy. Turning to the forward error, one approach is to evaluate the forward error bound from Theorem 7.2 or Theorem 7.4, with equal to the corresponding backward error. Because x in (7.9) is unknown, we should use the modified bound (7.24) If we have a particular E and f in mind for backward error reasons, then it is natural to use them in (7.24). However, the size of the forward error bound varies with E and f, so it is natural to ask which choice minimizes the bound.
P ERTURBATION T HEORY
144
FOR
LINEAR S YSTEMS
Lemma 7.9. The upper bound in (7.23) is at least as large as the upper bound in (7.25) and is equal to it when Proof. First note that r = b (7.25). Now, for z > 0,
is a multiple of |r|. implies
with equality if z is a multiple of r. Taking z =
with equality when is preserved when
which implies
gives
is a multiple of |r|. The truth of this statement -norms are taken, so the result follows.
Since the bound (7.25) is obtained by taking absolute values in the equation it is clearly the smallest possible such bound subject to ignoring signs in A-1 and r. It is reasonable to ask why we do not take as our error bound. (Theoretically it is an exact bound!) The reason is that we cannot compute r or exactly. In place of r we compute and (7.26) Therefore a strict bound. and one that should be used in practice in place of (7.25), is (7.27) This forward error bound is estimated and returned by LAPACK‘s xyyRFS routines. For details on how this is done without computing A -1, see Chapter 14. The LAPACK linear equation solvers estimate only one condition number: the standard condition number κ1 (A) (or, rather, its reciprocal, referred to as rcond), which is returned by the xyyCON routines.
7.8. Perturbation Theory by Calculus The perturbation results in this book are all derived algebraically, without any use of derivatives. Calculus can also be used to derive perturbation bounds, often in a straight forward fashion.
7.9 N OTES
AND
R EFERENCES
145
As a simple example, consider a linear system A(t)z(t) = b(t), where and x(t), b(t) are assumed to be continuously differentiable functions of t. Differentiating gives
or, dropping the t arguments,
Taking norms, we obtain
This bound shows that κ(A) is a key quantity in measuring the sensitivity of a linear system. A componentwise bound could have been obtained just as easily. We normally express perturbations of the data in the form To use the calculus framework we can take A(0) as the original matrix A and write but the perturbation bound then becomes a first-order one. The calculus technique is a useful addition to the armoury of the error analyst (it is used by Golub and Van Loan [470, 1989], for example), but the algebraic approach is preferable for deriving rigorous perturbation bounds of the standard forms.
7.9. Notes and References This chapter draws on the survey paper Higham [558, 1994]. Theorem 7.3 is due to Oettli and Prager [802, 196 4], and predates the normwise backward error result Theorem 7.1 of Rigal and Gaches [873, 1967]. In addition to Theorem 7.1, Rigal and Gaches give a more general result based on norms of blocks that includes Theorems 7.3 and 7.1 as special cases. Theorem 7.1 is also obtained by Kovarik [672, 1976]. Theorems 7.1 and 7.3 both remain valid when A is rectangular. Componentwise backward error for rectangular A was considered by Oettli, Prager, and Wilkinson [803, 1965], but their results are subsumed by those of Oettli and Prager [802, 1964] and Rigal and Gaches [873, 1967]. For a linear system Ax = b subject to componentwise perturbations, Oettli [801, 1965] shows how linear programming can be used to obtain bounds on the components of x when all solutions of the perturbed system lie in the same orthant. Cope and Rust [244, 1979] extend this approach by showing, in general, how to bound all the solutions that lie in a given orthant. This type
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P ERTURBATIO N T HEORY
FOR
LINEAR S YSTEMS
of analysis can also be found in the book by Kuperman [681, 1971], which includes an independent derivation of Theorem 7.3. See also Hartfiel [505, 1 9 8 0 ]. Theorem 7.4 is a straightforward generalization of a result of Skeel [919, 1979 , Thms. 2.1 and 2.2]. It is clear from Bauer’s comments in [80, 19 66] that the bound (7.9), with E = |A| and f = |b|, was known to him, though he does not state the bound. This is the earliest reference we know in which componentwise analysis is used to derive forward perturbation bounds. Theorem 7.8 is from Bauer [79, 1963]. Bauer actually states that equality holds in (7.21) for any A, but his proof of equality is valid only when |A - 1||A| and |A||A- 1| have positive Perron vectors. Businger [168, 1968] proves that a sufficient condition for the irreducibility condition of Theorem 7.8 to hold (which, of course, implies the positivity of the Perron vectors) is that there do not exist permutations P and Q such that PAQ is in block triangular form. Theorem 7.7 is from Stewart and Sun [954, 1990, Thm. 4.3.5]. Further results on scaling to minimize the condition number κ(A) are given by Forsythe and Straus [386, 1955], Bauer [81, 1969], Golub and Varah [465, 1974], McCarthy and Strang [742, 1974], Shapiro [913, 1982]. [914, 1985], [915, 1991], and Watson [1067, 1991]. Chan and Foulser [193, 1988] introduce the idea of “effective conditioning” for linear systems, which takes into account the projections of b onto the range space of A. See Problem 7.5, and for an application to partial differential equations see Christiansen and Hansen [208, 1994]. For an example of how definitions of numerical stability for linear equation solvers can be extended to incorporate structure in the problem, see Bunch [162, 1987]. An interesting application of linear system perturbation analysis is to Markov chains. A discrete-time Markov chain can be represented by a square matrix P, where pij is the probability of a transition from state i to state j . Since state i must lead to some other state. and these conditions can be writ ten in matrix vector form as Pe = e.
(7.28)
A nonnegative matrix satisfying (7.28) is called a stochastic matrix. The initial state of the Markov chain can be defined by a vector zT , where zi , denotes the probability that the ith state of the chain is occupied. Then the state of the chain at the next time unit is given by zTP. The steady state or stationary vector of the chain is given by
An important question is the sensitivity of the individual components of the steady-state vector to perturbations in P. This is investigated, for example.
147
PROBLEMS
by Ipsen and Meyer [605, 1994], who measure the perturbation matrix normwise, and by O’Cinneide [800, 1993], who measures the perturbation matrix componentwise. For a matrix-oriented development of Markov chain theory see Berman and Plemmons [94, 1994]. It is possible to develop probabilistic perturbation theory for linear systems and other problems by making assumptions about the statistical distribution of the perturbations. We do not consider this approach here (though see Problem 7.13), but refer the interested reader to the papers by Fletcher [376, 1985], Stewart [948, 1990], and Weiss, Wasilkowski, Wozniakowski, and Shub [1073, 19 86].
Problems 7.1. Under the conditions of Theorem 7.4, show that
Hence derive a first-order bound for |xi - yi |/|xi |. 7.2. Let Ax = b, where Show that for any vector y and any subordinate matrix norm,
where the residual r = b - Ay. Interpret this result. 7.3. Prove (7.13) and deduce (7.12). 7.4. Let where D = cond(H)
1 if E = |A| or f = |b|. Derive the corresponding normwise condition number in which the constraints are ||∆A||2 < and ||∆b|| 2 < 7.9. (Bauer [79, 1963]) Let A, B, C positive elements then
(a) Prove that if B and C have
where = {diag(d i ) : di > 0, i = 1:n}. (Hint: consider D 1 = diag(x1 ) - 1 and D 2 = diag(Cx1), where x1 > 0 is a right Perron vector of BC: BCx1 = p (B C) x1 . ) (b) Deduce that if |A| and |A - 1 | have positive entries, then
(c) Show that for any nonsingular A,
(d) Strengthen (b) by showing that for any nonsingular A such that |A||A- 1 | and |A- 1 ||A|| are irreducible,
PROBLEMS (e) What can you deduce about 2-norms?
149 for the 1- and
7.10. (Bauer [80, 1966, p. 413], Rohn [876, 1989]) We can modify the definition of µE(A) in (7.22) by measuring ∆X componentwise relative to X, giving
7.11. Let be symmetric and let y be an approximate solution to Ax = b. If y has a small backward error, so that y solves a nearby system. does it follow that y solves a nearby symmetric system? This problem answers the question for both the normwise and componentwise relative backward errors. (a) (Bunch, Demmel, and Van Loan [163, 1989]) Show that if (A+G)y = b then there exists H = HT such that (A + H)y = b with ||H|| 2 < ||G||2 and ||H|| F < (This result does not require A = AT.) (b) (Smoktunowicz [930, 1995]) Show that if A is symmetric and diagonally then there exists H = HT such dominant and (A + G)y = b with |G| < that (A + H)y = b with |H| < (For a general symmetric A there may not exist such an H, as is easily shown by a 2 × 2 example [527, 1992].) (c) (Smoktunowicz [930, 1995]) Show that if A is symmetric positive definite and (A + G)y = b with |G| < then there exists H = H T such that (A + H)y = b with |H| < 7.12. Suppose that has wi nonzeros in its ith row, i = 1:n. Show that the inequality (7.27) can be replaced by
where This bound is potentially much smaller than (7.27) for large, sparse matrices. 7.13. (D. J. Higham, after Fletcher [376, 198 5]) Suppose the nonsingular, square matrix A is perturbed to A + ∆A and b to b + ∆b. Then, to first order, the solution of Ax = b is perturbed to x + ∆x, where
P ERTURBATION T HEORY
150
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Suppose that the perturbations have the form
where the and δi are independent random variables, each having zero mean and variance σ2. (As usual, the matrix E and vector f represent fixed tolerances.) Let ε denote the expected value. (a) Show that
where square brackets denote the operation of elementwise squaring: [B] ij
=
(b) Hence explain why
may be regarded as an “expected condition number” for the linear system Ax = b. (c) For the case where eij ||A|| 2 and fj ||b|| 2. compare condexp(A, x) with the “worst-case” condition number κA , b (A, x) for the 2-norm. 7.14. (Horn and Johnson [581,
1991,
p. 331]) Prove that for any nonsingular
where is the Hadamard product (A B = (aijbij)) and is defined as in B||2 < ||A||2 ||B||2.) Discuss the Problem 7.9. (Hint: use the inequality ||A attainability of this bound.
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Chapter 8
Triangular Systems
In the end there is left the coefficient of one unknown and the constant term. An elimination between this equation and one from the previous set that contains two unknowns yields an equation with the coefficient of another unknown and another constant term, etc. The quotient of the constant term by the unknown yields the value of the unknown in each case. -JOHN V. ATANASOFF, Computing Machine for the Solution of Large Systems of Linear Algebraic Equations (1940)
The solutions of triangular systems are usually computed to high accuracy. This fact . . . cannot be proved in genera/, for counter examples exist. However, it is true of many special kinds of triangular matrices and the phenomenon has been observed in many others. The practical consequences of this fact cannot be over-emphasized. -G. W. STEWART, Introduction to Matrix Computations (1973)
In practice one almost invariably finds that if L is ill-conditioned, so that then the computed solution of Lx = b (or the computed inverse) is far more accurate than [standard norm bounds] would suggest. -J. H. WILKINSON, Rounding Errors in Algebraic Processes (1963)
151
152
TRIANGULAR SYSTEM
Triangular systems play a fundamental role in matrix computations. Many methods are built on the idea of reducing a problem to the solution of one or more triangular systems, including virtually all direct methods for solving linear systems. On serial computers triangular systems are universally solved by the standard back and forward substitution algorithms. For parallel computation there are several alternative methods, one of which we analyse in §8.4. Backward error analysis for the substitution algorithms is straightforward and the conclusion is well known: the algorithms are extremely stable. The behaviour of the forward error, however, is intriguing, because the forward error is often surprisingly small --much smaller than we would predict from the normwise condition number κ, or, sometirnes, even the componentwise condition number cond. The quotes from Stewart and Wilkinson at the start of this chapter emphasize the high accuracy that is frequently observed in practice. The analysis we give in this chapter provides a partial explanation for the observed accuracy of the substitution algorithms. In particular, it reveals three important but nonobvious properties: • the accuracy of the computed solution from substitution depends strongly on the right-hand side: • a triangular matrix may be much more or less ill conditioned than its transpose; and • the use of pivoting in LU, QR, and Cholesky factorizations can greatly improve the conditioning of a resulting triangular system. As well as deriving backward and forward error bounds, we show how to compute upper and lower bounds for the inverse of a triangular matrix.
8.1. Backward Error Analysis Recall that for an upper triangular matrix be solved using the formula xi = components of x in order from last to first.
the system Ux = b can which yields the
Algorithm 8.1 (back substitution). Given a nonsingular upper triangular matrix this algorithm solves the system Ux = b. xn = bn /u n n for i = n - 1:-1:1 s = bi for j = i + 1:n s = s - uijxij
8.1 B ACKWARD E RROR A NALYSIS
153
end x i = s/u ii end We will not state the analogous algorithm for solving a lower triangular system, forward substitution. All the results below for back substitution have obvious analogues for forward substitution. Throughout this chapter T denotes a matrix that can be upper or lower triangular. To analyse the errors in substitution we need the following lemma. Lemma 8.2. Let y = metic according to
be evaluated in floating point arith-
s =c for i = 1:1k-1 s = s - ai bi end y = s/bk Then the computed
satisfies (8.1)
where |θi | < γi = iu/(1 - iu). Proof. Analysis very similar to that leading to (3.2) shows that satisfies
where
:=
The final division yields, using (2.5), so that, after dividing through by
we have
The result is obtained on invoking Lemma 3.1. Two remarks are in order. First, we chose the particular form of (8.1), in which c is not perturbed, in order to obtain a backward error result for Ux = b in which b is not perturbed. Second, we carefully kept track of the terms 1 + δi in the proof, so as to obtain the best possible constants. Direct application of the lemma to Algorithm 8.1 yields a backward error result.
TRIANGULAR SYSTEMS
154 Theorem 8.3. The computed solution
from Algorithm 8.1 satisfies
Theorem 8.3 holds only for the particular ordering of arithmetic operations used in Algorithm 8.1. A result that holds for any ordering is a consequence of the next lemma. Lemma 8.4. If y = is evaluated in floating point arithmetic, then. no matter what the order of evaluation,
where
for all i. If bk = 1, so that there is no division, then for all i.
Proof. The result is not hard to see after a little thought , but a formal proof is tedious to write down. Note that the ordering used in Lemma 8.2 is the one for which this lemma is least obvious! The last part of the lemma is useful when analysing unit lower triangular systems, and in various other contexts. Theorem 8.5. Let the triangular system TX = b, where is nonsingular, be solved by substitution, with any ordering. Then the computed solution satisfies
In technical terms, this result says that has a tiny componentwise relative backward error. In other words, the backward error is about as small as we could possibly hope. In most of the remaining error analyses in this book, we will derive results that, like the one in Theorem 8.5, do not depend on the ordering of the arithmetic operations. Results of this type are more general, usually no less informative. and easier to derive, than ones that depend on the ordering. However, it is important to realise that the actual error does depend on the ordering, possibly strongly so for certain data. This point is clear from Chapter 4 on summation.
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8.2 F ORWARD E RROR A NALYSIS
8.2. Forward Error Analysis From Theorems 8.5 and 7.4 there follows the forward error bound
where
This bound can, of course, be arbitrarily smaller than the corresponding bound involving for the reasons explained in Chapter 7. For further insight, note that, in terms of the traditional condition number, κ(T), ill conditioning of a triangular matrix stems from two possible sources: variation in the size of the diagonal elements and rows with off-diagonal elements which are large relative to the diagonal elements. Significantly, because of its row scaling invariance, cond(T , x) is susceptible onlyto the second source. Despite its pleasing properties, cond(T , x) can be arbitrarily large. This is illustrated by the upper triangular matrix (8.2) for which (8.3) We have cond(U(a),e) = cond(U( a)) 2an-1 as a Therefore WC cannot assert that all triangular systems are solved to high accuracy. Nevertheless, for any T there is always at least one system for which high accuracy is obtained: the system TX = e1 if T is upper triangular, or TX = en if T is lower triangular. In both cases cond(T, x) = 1, and the solution comprises the computation of just a single scalar reciprocal. To gain further insight we consider special classes of triangular matrices, beginning with one produced by certain standard factorizations with pivoting. In all the results below, the triangular matrices are assumed to be n × n and nonsingular, and is the computed solution from substitution. Lemma 8.6. Suppose the upper triangular matrix
satisfies (8.4)
Then the unit upper triangular matrix W = |U- 1 ||U| satisfies wij < 2 j-i for all j > i.
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156
Proof. We can write W = |V- 1 ||V| where V = D -1 U and D = diag(u ii ). The matrix V is unit upper triangular with) |υij| < 1, and it is easy to show that |(V - 1 ) ji| < 2 j - i -1 for j > i. Thus. for j > i,
Theorem 8.7. Under the conditions of Lemma 8.6, the computed solution to Ux = b obtained by substitution satisfies
Proof. From Theorem 8.5 we have
Using Lemma 8.6 we obtain
Lemma 8.6 shows that for matrices satisfying (8.4), cond(T) is bounded for fixed n, no matter how large κ(T). The bounds for in Theorem 8.7, although large if n is large and i is small. decay exponentially with increasing thus, later components of x are always computed to high accuracy relative to the elements already computed. Analogues of Lemma 8.6 and Theorem 8.7 hold for lower triangular L satisfying |lii | > |lij| for all j < i. (8.5) Note, however, that if the upper triangular matrix T satisfies (8.4) then TT does not necessarily satisfy (8.5). In fact, cond(T T) can he arbitrarily large, as shown by the example
An important conclusion is that a triangular system Tx = b can be much more or less ill conditioned than the system TTy = c, even if T satisfies (8.4). Theorem 8.7, or its lower triangular analogue, is applicable to
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8.2 F ORWARD E RROR A NALYSIS
• the lower triangular matrices from Gaussian elimination with partial pivoting or complete pivoting; • the upper triangular matrices from Gaussian elimination with complete pivoting; • the upper triangular matrices from the Cholesky and QR factorizations with complete pivoting and column pivoting, respectively. Next, we consider triangular T satisfying tii > 0, tij < 0 for all i
j.
It is easy to see that such a matrix has an inverse with nonnegative elements, and hence is an hi-matrix (for definitions of an hi-matrix see Appendix B). Associated with any square matrix A is the comparison matrix: (8.6) For any nonsingular triangular T, M(T) is an M-matrix. Furthermore, it is easy to show that |T -1| < M(T)-1 (see Theorem 8.11). The following result shows that among all matrices R such that |R| = |T|, R = M(T) is the one that maximizes cond(R , x ). Lemma 8.8. For any triangular T,
Proof. The inequality follows from |T- 1 | < M(T)-1, together with |T| = |M(T)|. Since M(T)-1 > 0, we have |M(T)- 1 ||M(T)| = M(T) -1 (2diag(|tii| ) - M(T)) = 2M(T) -1
diag(|tii |) - I,
which yields the equality. If T=M(T) has unit diagonal then, using Lemma 8.8,
This means, for example, that the system U(1)x = b (see (8.2)), where x = e, is about as ill conditioned with respect to componentwise relative perturbations in U(1) as it is with respect to normwise perturbations in U(1).
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T RIANGULAR S YSTEMS
The next result gives a forward error bound for substitution that is proved directly, without reference to the backward error result Theorem 8.5 (indeed. it cannot be obtained from that result!). The bound can be very weak, because ||M(T) - 1 || can be arbitrarily larger than ||T- 1|| (see Problem 8.2), but it yields a pleasing bound in the special case described in the corollary. Theorem 8.9. The computed solution obtained from substitution applied to the triangular system Tx = b of order n satisfies
Proof. Without loss of generality, suppose T = L, is lower triangular. The proof is by induction on the components of x. The result clearly holds for the first component. Assume that it holds for the first n - 1 components. An analogue of Lemma 8.3 shows that
where gives
for all j. Subtracting from lnnxn =
so that (8.7) Write
Then the inductive assumption can be written as which implies Hence (8.7) gives
8.3 BOUNDS
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INVERSE
159
Corollary 8.10. The computed solution obtained from substitution applied to the triangular system Tx = b of order n, where T = M(T) and b > 0, satisfies Corollary 8.10 shows that, when T is an M-matrix and the right-hand side is nonnegative, the solution is obtained to high relative accuracy in every component. The reason for the high accuracy is that for such a system there are no subtractions of like-signed numbers, so that each xi is computed as a sum of nonnegative quantities. A consequence of the corollary is that the inverse of a triangular M-matrix can be computed to high relative accuracy. Triangular systems of the type in Corollary 8.10 occur in linear equations obtained by discretizing certain elliptic partial differential equations, such as the Poisson equation on a rectangle, with zero boundary conditions and a positive forcing function: these problems yield symmetric positive definite M-matrices, and the LU factors of an M-matrix are themselves M-matrices. Such systems also occur when evaluating the bounds of the next section. 8.3. Bounds for the Inverse In this section we describe bounds for the inverse of a triangular matrix and show how they can be used to bound condition numbers. All the bounds in this section have the property that they depend only on the absolute values of the elements of the matrix. The norm estimation methods of Chapter 14, on the other hand, do take account of the signs of the elements. The bounds are all based on the following theorem, whose easy proof we omit. Theorem 8.11. If U is a nonsingular upper triangular matrix then |U- 1 | < M(U)-1 < W(U)-1 < Z(U)- 1 , where the upper triangular matrices W(U) and Z(U) are defined as follows:
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T RIANGULAR S YSTEMS
where a = mink |ukk|, β = maxi<j
|uij|/|uii|.
Theorem 8.11 is a special case of results in the theory of Ill-matrices. For more general results couched in terms of matrix minorants and diagonal dominance, respectively, see Dahlquist [261, 1983] and Varga [1051, 1976]; see also Householder [587, 1964, Exercise 15, p. 58]. An obvious implication of the theorem is that for any vector z and any absolute norm
By taking z = |U|e, z = |U||x|, and z = e, respectively, we obtain upper The cost of computing these bounds for cond(U), cond(U,x), and bounds is just the cost of solving a triangular system with coefficient matrix M(U), W(U), or Z(U), which is easily seen to be O(n2). O(n), and O(1) flops, respectively. By comparison, computing any of these condition numbers exactly costs O(n3) flops. As an example, here is how to compute an upper bound for in n 2 flops. Algorithm 8.12. Given a nonsingular upper triangular matrix this algorithm computes µ = y n = 1 /|unn| for i = n - 1:-1:1 s=1 for j = i + 1:n s = s + |uij|yj end y i = y i /|uii| end µ= How good are these upper bounds? We know from Problem 8.2 that the ratio ||M(T)- 1 ||/||T - 1 || can be arbitrarily large, therefore any of the upper bounds can be arbitrarily poor. However, with suitable assumptions on T, more can be said. It is easy to show that if T is bidiagonal then |T- 1 | = M(T)- 1 . Since a bidiagonal system can be solved in O(n) flops, it follows that the three condition numbers of interest can each be computed exactly in O(n) flops when T is bidiagonal. As in the previous section, triangular matrices that result from a pivoting strategy also lead to a special result.
8.3 BOUNDS
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161
Theorem 8.13. Suppose the upper triangular matrix
satisfies
|uii | > |uij | for all j > i. Then, for the 1-, 2-, and
-norms,
(8.8) Proof. The left-hand inequality is trivial. The right-hand inequality follows from the expression (see Problem 8.5), together with ||A||2 < The inequalities from the second on in (8.8) are all equalities for the matrix with u ii = 1 and u ij = -1 (j > i). The question arises of whether equality is possible for the upper triangular matrices arising from QR factorization with column pivoting, which satisfy the inequalities (see Problem 18.5) (8.9) That equality is possible is shown by the parametrized matrix of Kahan [626, 19 66]
(8.10)
where c = cos(θ), s = sin(θ). It is easily verified that Un (θ) satisfies the inequalities (8.9)-as equalities, in fact. Prom (8.3), Un ( θ )-1 = (βi j ) is given by
Thus as and hence, for small θ,
where
It can also be verified that the matrix defined by (-1) j-i|uij| satisfies, for small θ, while 2 n - 1 /|unn|. Hence the upper bounds for ||U - 1 || can all be too big by a factor of order 2n - 1 .
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8.4. A Parallel Fan-In Algorithm Substitution is not the only way to solve a triangular system. In this section we describe a different approach that has been suggested for parallel computation. Any lower triangular matrix can be factorized L = L1 L2 . . . Ln , where Lk differs from the identity matrix only in the kth column:
(8.11)
The solution to a linear system Lx = b may therefore be expressed as (8.12) where M i = When evaluated in the natural right-to-left order, this formula yields a trivial variation of a column-oriented version of substitution. The fan-in algorithm evaluates the product (8.12) in [log(n + 1)] steps by the fan-in operation (which is the operation used in pairwise summation: see §4.1). For example, for n = 7 the calculation is specified by
where all the products appearing within a particular size of parenthesis can be evaluated in parallel. III general. the evaluation can be expressed as a binary tree of depth [log(n + 1)] + 1, with products M1b and Mi Mi-1 (i = 3, 5,. . . , 2[(n - 1)/2] + 1) at the top level and a single product yielding x at the bottom level. This algorithm was proposed and analysed by Sameh and Brent [889, 1977], who show that it can be implemented in time steps on processors. The algorithm requires about n 3 /10 operations and thus is of no interest for serial computation. Some pertinent comments on the practical significance of log n terms in complexity results are given by Edelman [341, 1993]. To derive an error bound while avoiding complicated notation that obscures the simplicity of the analysis, we take n = 7. The result we obtain is easily seen to be valid for all n. We will not be concerned with the precise values of constants, so we write cn for a constant depending on n . We assume that the inverses Mi = are formed exactly, because the errors in forming them affect only the constants. From the error analysis of matrix-vector and matrix-matrix multiplication (§3.5), we find that the computed solution satisfies (8.13)
8.4 A P ARALLEL FAN- I N A LGORITHM
163
where
Premultiplying (8.13) on the left by L, we find that the residual r = is a sum of terms of the form
All these terms share the same upper bound, which we derive for just one of them. For j = 5, k =
where we have used the property that, for any The overall residual bound is therefore of the form (8.14) or, on taking norms, (8.15) By considering the binary tree associated with the fan-in algorithm, and using the fact that the matrices at the ith level of the tree have at most 2i - 1 nontrivial columns, it is easy to see that we can take dn = an log n , where a is a constant of order 1. It is not hard to find numerical examples where the bound in (8.15) is approximately attained (for dn = 1) and greatly exceeds which is the magnitude required for normwise backward stability. One way to construct such examples is to use direct search (see Chapter 24). The key fact revealed by (8.15) is that the fan-in algorithm is only conditionally stable. In particular, the algorithm is normwise backward stable if L is well conditioned. A special case in which (8.14) simplifies is when L is an M matrix and b > 0: Problem 8.4 shows that in this case |L - 1 ||L||x| < (2n-1)|x|, so (8.14) yields < (2n - 1)2 dnu|L||x| + O(u 2 ), and we have componentwise backward stability (to first order).
164
T RIANGULAR S YSTEMS We can obtain from (8.15) the result (8.16)
which was proved by Sameh and Brent (889, 1977] (with a n = ¼n 2 log n + O(n log n)). However, (8.16) is a much weaker bound than (8.14) and (8.15). In particular, a diagonal scaling Lx = b (where Dj is diagonal) leaves (8.14) (and, to a lesser extent, (8.15)) essentially unchanged, but can change the bound (8.16) by an arbitrary amount. A forward error bound can be obtained directly from (8.13). We find that
(8.17) where M(L) is the comparison matrix (a bound of the same form as that in Theorem 8.9 for substitution-see the Notes and References and Problem 8.10). which can be weakened to (8.18) We also have the bound (8.19) which is an immediate consequence of (8.14). Either bound in (8.18) and (8.19) can be arbitrarily larger than the other, for fixed n . An example where (8.19) is the better bound (for large n) is provided by the matrix with lij , 1, for which |L- 1 ||L| has maximum element 2 and M(L)- 1 |L| has maximum element 2n - 1 .
8.5. Notes and References Section 8.2 is based on Higham [538, 198 9]. Many of the results presented in §§8.2 and 8.3 have their origin in the work of Wilkinson. Indeed, these sections are effectively a unification and extension of Wilkinson’s results in [1085, 1961], [1088, 1963], [1089, 1965]. Classic references for Theorems 8.3 and 8.5 are Wilkinson [1085, 196 1, p. 294], [1088, 1963, pp. 100-102], Forsythe and Moler [396, 1967, §21], and Stewart [941, 1973, pp. 150, 408-410]. Analogues of Theorem 8.7 and Corollary 8.10 for matrix inversion are proved by Wilkinson in [1085, 1961, pp. 322 323], and Corollary 8.10 itself is proved in [1089, 1965, pp. 250-251].
8.5 N OTES
AND
R EFERENCES
165
A result of the form of Theorem 8.9 holds for any triangular system solver that does not rely on algebraic cancellation-in particular, for the fan-in algorithm, as already seen in (8.17). See Problem 8.10 for a more precise formulation of this general result. The bounds in §8.3 have been investigated by various authors. The unified presentation given here is based on Higham [534, 1987]. Karasalo [642, 1974] derives an O(n) flops algorithm for computing ||M(T) - 1 ||F. Manteuffel [726, 1981] derives the first two inequalities in Theorem 8.11, and Algorithm 8.12. A different derivation of the equations in Algorithm 8.12 is given by Jennings [613, 1982, §9]. The formulae given in Problem 8.5 are derived directly as upper bounds for by Lemeire [699, 1975]. That can be computed in O(n) flops when B is bidiagonal, as was first pointed out by Higham [531, 1986]. Demmel and Kahan [296, 1990] derive an estimate for the smallest singular value σmin of a bidiagonal matrix B by using the inequality where They compute in O(n) flops as
Section 8.4 is adapted from Higham [560, 1995], in which error analysis is given for several parallel methods for solving triangular systems. The fan-in method is topical because the fan-in operation is a special case of the parallel prefix operation and several fundamental computations in linear algebra are amenable to a parallel prefix-based implementation, as discussed by Demmel [287, 1992], [288, 1993]. (For a particularly clear explanation of the parallel prefix operation see the textbook by Buchanan and Turner [154, 1992, §13.21.) The important open question of the stability of the parallel prefix implementation of Sturm sequence evaluation for the symmetric tridiagonal eigenproblem has recently been answered by Mathias [734, 1995]. Mathias shows that for positive definite matrices the relative error in a computed minor can be as large as a multiple of where is the smallest eigenvalue of the matrix; the corresponding bound for serial evaluation involves The analogy with (8.19), where we also see a condition cubing effect, is intriguing. Higham and Pothen [568, 1994] analyse the stability of the “partitioned inverse method” for parallel solution of sparse triangular systems with many right-hand sides. This method has been studied by several authors in the 1990s; see Alvarado, Pothen, and Schreiber [13, 1993] and the references therein. The idea of the method is to factor a sparse triangular matrix as L = L 1 L 2 . . . Ln = G 1 G 2 . . . Gm , where each Gi is a product of consecutive Lj terms and 1 < m < n, with m as small as possible subject to the Gi being sparse. Then the solution to Lx = b is evaluated as
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166
where each is formed explicitly and the product is evaluated from right to left. The advantage of this approach is that x can be computed in m serial steps of parallel matrix-vector multiplication.
8.5.1. LAPACK Computational routine xTRTRS solves a triangular system with multiple righthand sides: xTBTRS is an analogue for banded triangular matrices. There is no driver routine for triangular systems.
Problems Before you start an exercise session, make sure you have a g/ass of water and a mat or towel nearby. -MARIE HELVIN, Mode/ Tips for a Healthy Future (1994)
8.1. Show that under the no-guard-digit model (2.6). Lemma 8.2 remains true if (8.1) is changed to
and that the corresponding modification of Theorem 8.5 has
8.2. Show that for a triangular matrix T the ratio ||M(T) - 1||/||T - 1 || can be arbitrarily large. 8.3. Suppose the unit upper triangular matrix satisfies |uij| < 1 for j > i. By using Theorem 8.9. show that the computed solution from substitution on Ux = b satisfies
Compare with the result of applying Theorem 8.7. 8.4. Let be triangular and suppose T = M(T) and Tx = b > 0. Show that |T- 1||T||x| < (2n-1)|x|, and hence that cond(T, x) < 2n-1. This result show that a triangular M-matrix system with a nonnegative righthand side is very well conditioned with respect to componentwise relative perturbations, irrespective of the size of κ(T) (and so leads to an alternative proof of Corollary 8.10). 8.5. Show that for a triangular both the l- and -norms (a and β are defined in Theorem 8.11).
for
P ROBLEMS
167
8.6. Write detailed algorithms for efficiently computing
and
8.7. Bounds from diagonal dominance. (a) Prove the following result (Ahlberg and Nilson [8, 1963], Varah [1049, 1975]): if (not necessarily triangular) satisfies
(that is, A is strictly diagonally dominant by rows), then
(b) Hence show that (Varga [1051,
1976])
if
satisfies
for some positive diagonal matrix D = diag(di ) (that is, AD is strictly diagonally dominant by rows), then
(c) Use part (b) to provide another derivation of the upper bound 8.8. (a) Let be nonsingular. For a given i and j, determine, if possible, a ij such that A + is singular. Where is the “best” place to perturb A to make it singular? (b) Let T = U(1) in (8.2), so that, for example,
Show that Tn is made singular by subtracting 22-n from a certain element of Tn. 8.9. (Zha [1127, 1993]) Show that if c and s are nonnegative (with c2 + s2 = 1) then the Kahan matrix Un (θ) in (8.10) has as its second smallest singular value. (That there should be such an explicit formula is surprising; none is known for the smallest singular value.)
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8.10. Consider a method for solving triangular systems Tx = b that computes xi = fi (T, b) where, for all i, fi is a multivariate rational function in which the only divisions are by diagonal elements of L and such that when T = M(T) and b > 0 there are no subtractions in the evaluation of fi . Show that a bound holds of the form in Theorem 8.9, namely, for (8.20) Give an example of a triangular system solver for which (8.20) is not satisfied.
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Chapter 9
LU Factorization and Linear Equations
It appears that Gauss and Doolittle applied the method only to symmetric equations. More recent authors, for example, Aitken, Banachiewicz, Dwyer, and Crout . . . have emphasized the use of the method, or variations of it, in connection with non-symmetric problems . . . Banachiewicz . . . saw the point . . . that the basic problem is really one of matrix factorization, or “decomposition” as he called it. -PAUL S. DWYER, Linear Computations (1951) Intolerable pivot-growth [with partial pivoting] is a phenomenon that happens only to numerical analysts who are looking for that phenomenon. -WILLIAM M. KAHAN, Numerical Linear Algebra (1966) By 1949 the major components of the Pilot ACE were complete and undergoing trials . . . During 1951 a programme for solving simultaneous linear algebraic equations was used for the first time. 26th June, 1951 was a landmark in the history of the machine, for on that day it first rivalled alternative computing methods by yielding by 3 p.m. the solution to a set of 17 equations submitted the same morning. -MICHAEL WOODGER, The History and Present Use of Digital Computers at the National Physical Laboratory (1958). The closer one looks, the more subtle and remarkable Gaussian elimination appears. -LLOYD N. TREFETHEN, Three Mysteries of Gaussian Elimination (1985)
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9.1. Gaussian Elimination We begin by giving a traditional description of Gaussian elimination (GE) for solving a linear system Ax = b, where is nonsingular. The strategy of GE is to reduce a problem we can’t solve (a full linear system) to one that we can (a triangular system), using elementary row operations. There are n - 1 stages, beginning with A(1) := A, b (1) := b, and finishing with the upper triangular system A(n)x = b (n) . At the kth stage we have converted the original system to A(k)x = b(k),
with upper triangular. The purpose of the kth stage of the elimination is to zero the elements below the diagonal in the kth column of A ( k ). This is accomplished by the operations
At the end of the (n - 1)st where the multipliers stage we have the upper triangular system A(n)x = b (n) , which is solved by back substitution. For an n × n matrix, GE requires 2n 3/3 flops. There are two problems with the method as described. First. there is a breakdown with division by zero if Second, if we are working in finite precision and some multiplier mik is large, then there is a possible loss of significance: in the subtraction low-order digits of could be lost. Losing these digits could correspond to making a relatively large change to the original matrix A. The simplest example of this phenomenon is for the matrix here. and if which would be the exact answer if we changed a22 from 1 to 0. These observations motivate the strategy of partial pivoting. At the start of the kth stage, the kth and rth rows are interchanged, where
Partial pivoting ensures that the multipliers are nicely bounded:
A more expensive pivoting strategy, which interchanges both rows and columns, is complete pivoting.
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At the start of the kth stage, rows k and r and columns k and s are interchanged, where Note that this requires O(n3 ) comparisons in total, compared with O(n2 ) for partial pivoting. Because of the searching overhead, and because partial pivoting works so well, complete pivoting is rarely used in practice. Much insight into GE is obtained by expressing it in matrix notation. We can write
The matrix Mk can be expressed compactly as Mk = where ek is for i < k. To invert Mk, just flip the signs the kth unit vector and of the multipliers: Overall,
and so
The conclusion is that GE computes an LU factorization of A : A = LU, where L is unit lower triangular and U is upper triangular. We introduce the shorthand notation Ak := A(1:k, 1:k). Theorem 9.1. There exists a unique LU factorization of if and only if Ak is nonsingular for k = 1:n - 1. If Ak is singular for some 1 < k < n - 1 then the factorization may exist, but if so it is not unique.
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Proof. Suppose Ak is nonsingular for k = 1:n - 1. The existence of an LU factorization can be proved by. examining the steps of GE. but a more elegant proof, which also gives uniqueness, can be obtained by an inductive bordering construct ion. Suppose Ak-1 has the unique LU factorization A k-1 = Lk- 1 U k-1 (this supposition clearly holds for k - 1 = 1). We look for a factorization
The equations to be satisfied are Lk- 1u = and a kk = 1 Tu + u k k . The matrices Lk-1 and Uk-1 are nonsingular, since 0 det(A k-1) = det(L k-1) det(U k-1). so the equations have a unique solution. completing the induction. We prove the converse, under the assumption that A is nonsingular; for the case A singular see Problem 9.1. Suppose an LU factorization exists. Then Ak = LkUk for k = 1:n, which gives det(A k) = det(U k) = u 11 . . . ukk.
(9.1)
Setting k = n we find that 0 det(A) = u 11 . . . unn, and hence det(A k) = u 11 . . . ukk 0, k = 1:n - 1. Examples that illustrate the last part of the theorem are which does not have an LU factorizawhich holds for any l, and tion. Visually, the condition of Theorem 9.1 is (for n = 5) that the indicated submatrices must be nonsingular:
From (9.1) follows the expression u kk = det(A k)/det(A k-l). In fact, all the elements of L and U can be expressed by determinant al formulae (see, e.g., Gantmacher [413, 1959, p. 35] or Householder [587, 1964. p. 11]): (9.2a) (9.2b)
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The effect of partial pivoting is easily seen by considering the case n = 4. We have
where, for example, For k = 1,2,3, is the same as Mk except the multipliers are interchanged. Hence, for n = 4, GE with partial pivoting (GEPP) applied to A is equivalent to GE without pivoting applied to the row-permuted matrix PA. This conclusion is true for any n: GEPP computes a factorization PA = LU. Similarly, GE with complete pivoting computes a factorization PAQ = LU, where P and Q are permutation matrices. Exploitation of the LU factorization streamlines both the error analysis and the practical solution of linear systems. Solution of Ax = b breaks into a factorization phase, PA = LU for partial pivoting (O(n 3 ) flops), and a substitution phase, where the triangular systems Ly = Pb, Ux = y are solved (O(n2 ) flops). If more than one system is to be solved with the same coefficient matrix but different right-hand sides, the factorization can be reused, with a consequent saving in work. Computing an LU factorization A = LU is equivalent to solving the equations
If these nonlinear equations are examined in the right order, they are easily solved. For added generality let (m > n) and consider an LU factorization with and (L is lower trapezoidal: lij = 0 for i < j). Suppose we know the first k - 1 columns of L and the first k - 1 rows of U. Setting lkk = 1, (9.3) (9.4) We can solve for the boxed elements in the kth row of U and then the kth column of L. This process is called Doolittle’s method. Algorithm 9.2 (Doolittle’s method). This algorithm computes an LU factorization A = where m > n (assuming the factorization exists), by Doolittle’s method.
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for k = 1: n for j = k :n (*) end for i = k + 1:m (**) end end Cost: n 2 (m - n/3) flops. Doolittle’s met hod is mathematically equivalent to GE without pivoting, for we have, in (9.3). (9.5) and similarly for (9.3). Had we chosen the normalization u ii 1, we would have obtained the Crout method. The Crout and Doolittle methods are well suited to calculations by hand or with a desk calculator, because they obviate the need to store the intermediate quantities They are also attractive when we can accumulate inner products in extended precision. It is straightforward to incorporate partial pivoting into Doolittle’s method (see. e.g.. Stewart [941, 1973, p. 138]). However, complete pivoting cannot be incorporated without changing the met hod.
9.2. Error Analysis The error analysis of GE is a combination of the error analyses of inner products and substitution. When this fact is realized. the analysis becomes straight forward. The key observation leading to this viewpoint is that all mathematically equivalent variants of GE satisfy a common error bound. To see why, first note the connection between standard GE, as we have described it, and the Doolittle met hod, as shown in (9.5). Whether the inner product in (9.5) is calculated as one operation. or whether its terms are calculated many operations apart, precisely the same rounding errors are sustained (assuming that extended precision accumulation of inner products is not used): all that changes is the moment when those rounding errors are committed. If we allow inner products to be reordered. so that, for example, the summation (*) in Algorithm 9.2 is calculated with the index i decreasing from k - 1 to 1, instead of increasing from 1 to k - 1. then the actual rounding errors are different but a common bound holds for all orderings. It suffices, then, to analyse the Doolittle method. It also suffices to analyse the met hod without pivoting, because GE with partial or complete pivoting is equivalent to GE without pivoting applied to a permuted matrix.
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The assignments (*) and (**) in Algorithm 9.2 are of the form y = ( c which is analysed in Lemma 8.4. Applying the lemma, we deduce that, no matter what the-ordering of the inner products, the computed matrices L and U satisfy (with lkk := 1)
These inequalities constitute a backward error result for LU factorization. Theorem 9.3. If GE applied to the computed LU factors
and
(m > n) runs to completion then satisfy (9.6)
With only a little more effort, a backward error result can be obtained for the solution of Ax = b. Theorem 9.4. Let and suppose GE produces computed LU factors and a computed solution to Ax = b. Then (9.7) Proof. From Theorem 9.3, Theorem 8.5, substitution produces
By and
satisfying
Thus
where
We need a constant 2γ n instead of Although it is not usually worth expending effort reducing constants in error bounds (see the Wilkinson quotation at the start of Chapter 10), we will optimize constants in this important case. Consideration of Lemma 8.4 shows that we actually have
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so that
using Lemma 3.3, which gives the required constant. How do we interpret Theorem 9.4? Ideally, we would like |∆ A| < u|A|, which corresponds to the uncertainty introduced by rounding the elements of A, but because each element of A undergoes up to n arithmetic operations we cannot expect better than a bound |∆ A| < cnu|A|, where cn is a constant of order n. Such a bound holds if and satisfy which certainly holds if and are nonnegative. because then (9.6) gives (9.8)
Substituting into (9.7). we obtain
This result says that has a small componentwise relative backward error. One class of matrices that has nonnegative LU factors is defined as follows. is totally positive (nonnegative) if the determinant of every square submatrix is positive (nonnegative). In particular, this definition requires that a ij and det(A) be positive or nonnegative. Some examples of totally nonnegative matrices are given in Chapter 26. If A is totally nonnegative then it has an LU factorization A = LU in which L and U are totally nonnegative. so that L > 0 and U > 0 (see Problem 9.6); moreover, the computed factors and are nonnegative for sufficiently small values of the unit roundoff u [273, 1977 ]. Inverses of totally nonnegative matrices also have the property that |A| = |L||U| (see Problem 9.7). Note that the property of a matrix or its inverse being totally nonnegative is generally destroyed under row permutations. Hence for totally- nonnegative matrices and their inverses it is best to use Gaussian elimination without pivoting. One important fact that follows from (9.6) and (9.7) is that the stability of GE is determined not by the size of the multipliers but by the size of the matrix This matrix can be small when the multipliers are large, and large when the multipliers are of order 1 (as we will see in the next section). To understand the stability of GE further we turn to norms. For GE without pivoting. the ratio || |L||U|||/||A|| can be arbitrarily large. For example, for the matrix the ratio is of order Assume then that partial pivoting is used. Then |lij | < 1 for all i > j, since the lij are the multipliers.
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And it is easy to show by induction that |uij| < 2 i -1 maxk < i |akj|. Hence, for partial pivoting, L is small and U is bounded relative to A. Traditionally, backward error analysis for GE is expressed in terms of the growth factor
which involves all the elements tion. Using the bound classic theorem. Theorem 9.5 (Wilkinson). Let oting produces a computed solution
(k = 1:n) that occur during the eliminap n maxi,j|aij| we obtain the following
and suppose GE with partial pivto Ax = b. Then (9.9)
We hasten to admit to using an illicit manoeuvre in the derivation of this theorem: we have used bounds for L and U that strictly are valid only for the exact L and U. We could instead have defined the growth factor in terms of the computed but then any bounds for the growth factor would involve the unit roundoff (similarly, we can only guarantee that < 1 + u). Our breach of correctness is harmless for the purposes to which we will put the theorem. The assumption in Theorem 9.5 that partial pivoting is used is not necessary: essentially the same result holds for GE without pivoting (see Problem 9.8). The normwise backward stability of GE with or without pivoting is therefore governed by the growth factor, to which we now turn our attention.
9.3. The Growth Factor It is easy to show that pn < 2 n -1 for partial pivoting. Wilkinson notes that this upper bound is achieved for matrices of the form illustrated for n = 4 by
For these matrices, no interchanges are required by partial pivoting, and there is exponential growth of elements in the last column of the reduced matrices. In fact, this is just one of a nontrivial class of matrices for which partial
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pivoting achieves maximal growth. When necessary in the rest of this chapter, we denote the growth factor for partial pivoting by and that for complete pivoting by Theorem 9.6 (Higham and Higham). All real n × n matrices A for which 2n -1 are of the form
where D = diag(±1). M is unit lower triangular with mij = -1 for i > j, T is an arbitrary nonsingular upper triangular matrix of order n -1, d = (1,2,4,,.... 2n -1)T, and a is a scalar such that a := |a1n| = maxi,j |aij|. Proof. GE with partial pivoting applied to a matrix A gives a factorization B := PA = LU, where P is a permutation matrix. It is easy to show that |uij | < 2i -1 maxr< i |brj|, with equality for i = s only if there is equality for i = 1:s - 1. Thus pn = 2n -1 implies that the last column of U has the form aDd, and also that |b1 n| = maxi,j |bij|. By considering the final column of B, and imposing the requirement that |lij| < 1, it is easy to show that the unit lower triangular matrix L must have the form L = DMD. It follows that at each stage of the reduction every multiplier is ±1; hence no interchanges are performed, that is, P = I. The only requirement on T is that it be nonsingular, for if t i i = 0 then the ith elimination stage would be skipped because of a zero pivot column and no growth would be produced on that stage. Note that by varying the elements mij (i > j) and the vector d in Theorem 9.6 we can construct matrices for which achieves any desired value between 1 and 2n - 1 . Despite the exist once of matrices for which pn is large with partial pivoting, the growth factor is almost invariably small in practice. For example, Wilkinson says “It is our experience that any substantial increase in size of elements of successive Ar is extremely uncommon even with partial pivoting No example which has arisen naturally has in my experience given an increase by a factor as large as 16” [1089, 1965, pp. 213-214]. Until recently. there were no reports in the literature of large growth factors being observed in practical applications. However, Wright [1116, 1993] has found a class of two-point boundary value problems that. when solved by the multiple shooting met hod, yield a linear system for which partial pivoting suffers exponential growth. The matrix is block lower bidiagonal, except for a nonzero block in the top right-hand corner. Furthermore, Foster [399, 1994] shows that a quadrature met hod for solving a practically occurring Volterra
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integral equation gives rise to linear systems for which partial pivoting again gives large growth factors. There exist some well-known matrices that give unusually large, but not exponential, growth. They can be found using the following theorem, which is applicable whatever the strategy for interchanging rows and columns in GE. Theorem 9.7 (Higham and Higham). Let be nonsingular and set a = max i,j |aij|, β = maxi,j |(A- 1 )ij |, and θ = (aβ)-1. Then θ < n, and for any permutation matrices P and Q such that PAQ has an LU factorization. the growth factor pn for GE without pivoting on PAQ satisfies p n > θ. Proof. The inequality θ < n follows from an LU factorization PAQ = LU computed by GE. We have
Consider
(9.10)
Hence
and the result follows.
Note that θ -1 = aβ satisfies Clearly, A has to be very well conditioned for the theorem to provide a lower bound θ near the maximum of n. We apply the theorem to three noncontrived matrices that appear in practical applications. (1) The matrix (9.11) is the symmetric, orthogonal eigenvector matrix for the second difference matrix (the tridiagonal matrix with typical row (-1, 2, -1)-see §26.5); it arises, for example, in the analysis of time series [19, 1971, §6.5]. Theorem 9.7 gives p n (S N) > (n + 1)/2. (2) A Hadamard matrix H n is an n × n matrix with elements h ij = ±1 and for which the rows of Hn are mutually orthogonal. Hadamard mat rices exist only for certain n; a necessary condition for their existence if n > 2 is that n is a multiple of 4. For more about Hadamard mat rices see Hall [494, 19 6 7 , Chap. 14], Wallis [1062, 1993 ], and Wallis, Street, and Wallis [1063, 1972 ]. We have = nI, and so Theorem 9.7 gives pn > n. (3) The next matrix is a complex Vandermonde matrix based on the roots of unity, which occurs in the evaluation of Fourier transforms (see §23.1): (9.12)
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Since Theorem 9.7 gives pn (Vn ) > n. Note that each of these matrices is orthogonal or unitary (to within a row scaling in the case of the Hadamard matrix), so it is not necessary to apply GE to them! This may explain why growth factors of order n for these matrices have not been reported in the literature as occurring in practice. To summarize, although there are practically occurring matrices for which partial pivoting yields a moderately large, or even exponentially large, growth factor, the growth factor is almost invariably found to be small. Explaining this fact remains one of the major unsolved problems in numerical analysis. The best attempt to date is by Trefethen and Schreiber [1019, 1990]. who develop a statistical model of the average growth fact or for partial pivoting and complete pivoting. Their model supports their empirical findings that for various distributions of random mat rices the average growth fact or (normalized by the standard deviation of the initial matrix elements) is close to n 2 / 3 for partial pivoting and n ½ for complete pivoting. Extensive experiments by Edelman suggest that for random matrices from the normal N(0, 1) distribution the unnormalized growth factor for partial pivoting grows like n ½ [345, 1 9 9 5 ]. We turn now to complete pivoting. Wilkinson [1085, 1961, pp. 282-285] showed that (9.13) and that this bound is not attainable. The bound is a much more slowly growing function than 2n -1, but can still be quite large (e.g., it is 3570 for n = 100). As for partial pivoting, in practice the growth factor is usually small. Wilkinson stated that “no matrix has been encountered in practice for which p 1 /pn was as large as 8” [1085, 1961, p. 285] and that “no matrix has yet been discovered for which f(r) > r” [1089, 1965, p. 213] (here. pi is the (n - i + 1)st pivot and f(r) Cryer [256, 1968] defined (9.14) The following results are known: • g(2) = 2 (trivial). • g(3) = 2¼; Tornheim [1012, • g (4) = 4; Cryer [256,
1968]
• g(5) < 5.005; Cohen [229,
1965]
and Cohen [229,
and Cohen [229, 1974].
1974].
1974].
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Tornheim [1012, 1965] (see also Cryer [256, 1968]) showed that > n for any n × n Hadamard matrix Hn (a bound which, as we saw above, holds for any form of pivoting). For n such that a Hadamard matrix does not exist, the best known lower bound is g(n) > = (n + 1)/2 (see (9.11)). Cryer [256, 1968] conjectured that for real matrices < n, with equality if and only if A is a Hadamard matrix. The conjecture < n became one of the most famous open problems in numerical analysis, and has been investigated by many mathematicians. The conjecture was finally shown to be false in 1991. Using a package LANCELOT [236, 1992] designed for large-scale nonlinear optimization, Gould [474, 1991] discovered a 13 × 13 matrix for which the growth factor is 13.0205 in IEEE double precision floating point arithmetic. Edelman subsequently showed, using the symbolic manipulation packages Mathematica and Maple, that a growth factor 13.02 can be achieved in exact arithmetic by making a small perturbation (of relative size 10-7) to one element of Gould’s matrix [338, 1992], [348, 1991]. A more striking counterex= 32.986341 [338, ample to the conjecture is a matrix of order 25 for which 1992]. Interesting problems remain, such as determining limn g(n)/n and evaluating for Hadamard matrices (see Problem 9.15). For complex matrices the maximum growth factor is at least n for any n, since > n (see (9.12)). The growth can exceed n, even for n = 3: Tornheim [1012, 1965] constructed the example
for which
= 3.079.
9.4. Special Matrices For matrices with certain special properties, more can be said about the behaviour of GE and, in particular, the size of the growth factor. As a first example, suppose is diagonally dominant by rows,
or diagonally dominant by columns, that is, A* is diagonally dominant by rows. Then GE without pivoting is perfectly stable. Theorem 9.8 (Wilkinson). If is diagonally dominant by rows or columns then A has an LU factorization without pivoting and the growth factor pn < 2. If A is diagonally dominant by columns then |lij| < 1 for all i and j in the LU factorization without pivoting (hence GEPP does not require any row interchanges).
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Figure 9.1. A banded matrix. Proof. The result follows immediately from the more general Theorems 12.5 and 12.6 for block diagonally dominant matrices. Note that for a matrix diagonally dominant by rows the multipliers can be arbitrarily large but, nevertheless, pn < 2, so GE is perfectly stable. A smaller bound for the growth factor also holds for an upper Hessenberg matrix. (H is upper Hessenberg if hij = 0 for i > j + 1.) Theorem 9.9 (Wilkinson). If
is upper Hessenberg then
< n.
Proof. The structure of an upper Hessenberg H means that at each stage of GEPP we just add a multiple of the pivot row to the next row (after possibly swapping these two rows). That < n is a consequence of the following claim, which is easily proved by induction: at the start of stage k , row k + 1 of the reduced matrix is the same as row k + 1 of the original matrix, and the pivot row has elements of modulus at most k times the largest element o f H. has lower bandwidth p and upper bandwidth q if aij = 0 A matrix for i > j + p and j > i + q; see Figure 9.1. It is well known that in an LU factorization of a banded matrix the factors inherit A's band structure: L has lower bandwidth p and U has upper bandwidth q. If partial pivoting is used then, in PA = LU. L has at most p + 1 nonzeros per column and U has upper bandwidth p + q. (For proofs of these properties see Golub and Van Loan [470, 1989, §4.3].) It is not hard to see that for a banded matrix, γ n in Theorem 9.3 can be replaced by γ m a x (p + 1 , q+1) and 2γ n in Theorem 9.4 can be replaced by γm a x (p + 1 , q+1) + γp + q + 1 . ’ The following result bounds the growth factor for partial pivoting on a banded matrix.
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Theorem 9.10 (Bohte). If has upper and lower bandwidths p then and this bound is almost attainable when n = 2p+1. Proof. See Bohte [131, 1975]. An example with n = 9 and p = 4 in which equality is almost attained is the matrix
where is an arbitrarily small positive number, which ensures that rows 1 and 5 are interchanged on the first stage of the elimination, this being the only row interchange required. Ignoring terms in the last column of U in PA = LU is [1, 1, 2, 4, 8, 16, 31, 60, 116]T and the growth factor is 116. A special case of Theorem 9.10 is the easily verified result that for a tridiagonal matrix, Hence GEPP achieves a small normwise backward error for tridiagonal matrices. In the next section we show that for several types of tridiagonal matrix GE without pivoting achieves a small componentwise backward error.
9.5. Tridiagonal Matrices Consider the nonsingular tridiagonal matrix
and assume that A has an LU factorization A = LU, where
(9.15)
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GE for computing L and U is described by the recurrence relations (9.16) For the computed quantities, we have
Hence
In matrix terms these bounds may be written as (9.17) If the LU factorization is used to solve a system Ax = b by forward and back substitution then it is straightforward to show that the computed solution satisfies (9.18) Combining (9.17) and (9.18) we have, overall, (9.19) The backward error result (9.19) applies to arbitrary nonsingular tridiagonal A having an LU factorization. We are interested in determining classes of tridiagonal A for which a bound of the form |∆ A| < g (u) |A| holds. Such a bound will hold if as noted in §9.2 (see (9.8)). Three classes of matrices for which holds for the exact L and U are identified in the following theorem. Theorem 9.11. Let be nonsingular and tridiagonal. If any of the following conditions hold then A has an LU factorization and |L||U| = |LU|: (a) A is symmetric positive definite; (b) A is totally nonnegative, or equivalently, L > 0 and U > 0; (c) A is an M-matrix, or equivalently, L and U have positive diagonal elements and nonpositive off-diagonal elements; (d) A is sign equivalent to a matrix B of type (a)-(c), that is, A = D 1 BD2 , where |D1 | = |D2 | = I.
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Proof. For (a), it is well known that a symmetric positive definite A has an LU factorization in which U = DL T , where D is diagonal with positive diagonal elements. Hence |L||U| = |L||D||LT| = |LDLT| = |LU|, where the middle equality requires a little thought. In (b) and (c) the equivalences, and the existence of an LU factorization, follow from known results on totally nonnegative matrices [258, 1976] and M-matrices [94, 1994]; |L||U| = |LU| is immediate from the sign properties of L and U. (d) is trivial. For diagonally dominant matrices, |L||U| is not equal to |LU| = |A|, but it cannot be much bigger. is nonsingular, tridiagonal, and diagTheorem 9.12. Suppose onally dominant by rows or columns, and let A have the LU factorization A = LU. Then |L||U| < 3|A|. Proof. If |i - j| = 1 then (|L||U|) ij = |aij|, so it suffices to consider the diagonal elements and show that (using the notation of (9.15)) |li ei-1 | + |ui | < 3|di |. The rest of the proof is for the case where A is diagonally dominant by rows; the proof for diagonal dominance by columns is similar. First, we claim that |ei | < |ui | or all i. The proof is by induction. For i = 1 the result is immediate, and if it is true for i - 1 then, from (9.16).
as required. Note that, similarly, |ui | < |di | + |ci |. Finally,
Theorem 9.13. If the nonsingular tridiagonal matrix A is of type (a)-(d) in Theorem 9.11, and if the unit roundoff u is sufficiently small, then GE for solving Ax = b succeeds and the computed solution satisfies
The same conclusion is true if A is diagonally dominant by rows or columns, with no restriction on u, provided the bound is multiplied by 3.
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Proof. If u is sufficiently small then for types (a)-(c) the diagonal elements of U will be positive, since as u 0. It is easy to see that for all i ensures that The argument is similar for type (d). The result therefore follows from (9.19) and (9.8). The last part is trivial. A corollary of Theorem 9.13 is that it is not necessary to pivot for the matrices specified in the theorem (and, indeed, pivoting could vitiate the result of the theorem). Note that large multipliers may occur under the conditions of the theorem, but they do not affect the stability. For example, consider the M-matrix
where The multiplier l32 is unbounded as but |L||U| = |A| and GE performs very stably, as Theorem 9.13 shows it must.
9.6. Historical Perspective GE was the first numerical algorithm to be subjected to rounding error analysis, so it is instructive to follow the development of the error analysis from its beginnings in the 1940s. In the 1940s there were three major papers giving error analyses of GE. Hotelling [583, 1943] presented a short forward error analysis of the LU factorization stage of GE. Under the assumptions that |aij| < 1 and |bi | < 1 for all i and j and that the pivots are all of modulus unity, Hotelling derives a bound containing a fact or 4 n -1 for the error in the elements of the reduced upper triangular system. Hotelling’s work was quoted approvingly by Bargmann. Montgomery, and von Newmann [55, 1946], who dismiss elimination met hods for the solution of a linear system Ax = b as being numerically unstable. Instead, they recommended computation of A-1 via the Newton Schulz iteration [908, 1933] (which was also discussed by Hotelling). In one paragraph they out line the alleged shortcomings of elimination methods as follows: In the elimination method a series of n compound operations is performed each of which depends on the preceding. An error at any stage affects all succeeding results and may become greatly magnified; this explains roughly why instability should be expected. It should be noticed that at each step a division is performed by a number whose size cannot be estimated in advance and which might the so small that any error in it would be greatly magnified by division. In fact such small divisors must occur if the determinant of the matrix is small and may occur even if it is not
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. . . Another reason to expect instability is that once the variable xn is obtained all the other variables are expressed in terms of it. As Wilkinson [1098, 1974, p. 354] notes of this paragraph, “almost every statement in it is either wrong or misleading”. Hotelling’s result led to general pessimism about the practical effectiveness of GE for solving large systems of equations. Three papers later in the same decade helped to restore confidence in GE. Goldstine [460, 1972, p. 290] says of his discussions with von Neumann: We did not feel it reasonable that so skilled a computer as Gauss would have fallen into the trap that Hotelling thought he had noted . . . Von Neumann remarked one day that even though errors may build up during one part of the computation, it was only relevant to ask how effective is the numerically obtained solution, not how close were some of the auxiliary numbers, calculated on the way to their correct counterparts. We sensed that at least for positive definite matrices the Gaussian procedure could be shown to be quite stable. von Neumann and Goldstine [1057, 1947] subsequently gave a long and difficult rigorous fixed-point error analysis for the inversion of a symmetric positive definite matrix A via GE. They showed that the computed inverse satisfies < 14.2n 2 uκ2 (A). Parlett [821, 1990] explains that “the joy of this result was getting a polynomial in n, and the pain was obtaining 14.2, a number that reflects little more than the exigencies of the analysis.” Wilkinson [1095, 1971] gives an interesting critique of von Neumann and Goldstine’s paper and points out that the residual bound could hardly be improved using modern error analysis techniques. In a later paper [462, 1951], Goldstine and von Neumann gave a probabilistic analysis, which Goldstine summarizes as showing that “under reasonable probabilistic assumptions the error estimates of the previous paper could be reduced from a proportionality of n 2 to n” [460, 1972, p. 291]. In his 1970 Turing Award Lecture [1096, 1971], Wilkinson recounts how in the early 1940s he solved a system of 12 linear equations on a desk calculator, obtaining a small residual. He goes on to describe a later experience: It happened that some time after my arrival [at the National Physical Laboratory in 1946], a system of 18 equations arrived in Mathematics Division and after talking around it for some time we finally decided to abandon theorizing and to solve it . . . The operation was manned by Fox, Goodwin, Turing, and me, and we decided on Gaussian elimination with complete pivoting . . . Again the system was mildly ill-conditioned, the last equation had a coefficient of order 10-4 (the original coefficients being of order unity) and
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the residuals were again of order 10-10 , that is of the size corresponding to the exact solution rounded to ten decimals. It is interesting that in connection with this example we subsequently performed one or two steps of what would now be called “iterative refinement,” and this convinced us that the first solution had had almost six correct figures. (Fox [403, 19 8 7 ] notes that the computation referred to in this quotation took about two weeks using desk computing equipment!) In a subsequent paper, Fox, Huskey, and Wilkinson [404, 1948] presented empirical evidence in support of GE, commenting that “in our practical experience on matrices of orders up to the twentieth, some of them very ill-conditioned, the errors were in fact quite small”. The experiences of Fox, Huskey, and Wilkinson prompted Turing to write a remarkable paper “Rounding-off errors in matrix processes” [1027, 1948]. In this paper, Turing made several import ant contributions. He formulated the LU (actually. the LDU) factorization of a matrix, proving the “if” part of Theorem 9.1 and showing that GE computes an LDU factorization. He introduced the term “condition number” and defined two matrix condition numbers, one of which is n - 1N(A )N( A -1), where N(A) = ||A||F, the “Ncondition number of A”. He used the word “preconditioning” to mean improving the condition of a system of linear equations (a term that did not come into popular use until the 1970s). He described iterative refinement for linear systems. He exploited backward error ideas, for example by noting that *‘the triangular resolution obtained is an exact resolution of a matrix A - S, where M(S) < ( M(S) := maxi,j |sij|. Finally, and perhaps most importantly, he analysed GEPP for general matrices and obtained a bound for that contains a term proportional to (By making a trivial change in the analysis, namely replacing A - 1 b by x, Turing’s bound can be made proportional only to Turing also showed that the factor 4n -1 in Hotelling’s bound can be improved to 2 n -1 and that still the bound is attained only in exceptional cases. In a review of Turing’s paper, Bodewig [129, 1949] described the error bounds as “impractical“ and advocated computing the residual of the computed solution and then determining “the exact correct ion by solving a new system.” That another researcher could miss the point of Turing’s analysis emphasizes how new the concept of rounding error analysis was in the 1940s. Table 9.1 shows the time for solution of linear systems by GE on some early computing devices. The performance of modern computers on two linear system benchmarks is summarized by Dongarra [312, 1995]; Dongarra’s report is regularly updated and can be obtained from netlib under the benchmark directory. Douglas [319, 1959] presented a forward error analysis for GE applied to
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Table 9.1. Times for solution of a linear system of order n. Machine Logarithm tables Desk computing equipment Harvard Mark 1 IBM 602 Calculating Punch Pilot ACE Pilot ACE’ ACE EDSAC 2 EDSAC 2 d
Year c. 1884 c. 1946 1947 1949 1951 1954 1958 1960 1960
Time n 7 weeks 29 a 2 weeks 18 45 minutes 10 4 hours 10 17 over 3 hours 1½ mins 30 5 seconds 30 4 seconds 31 100 7 minutes
Reference [952, 1994] [403, 1987] b [1053, 1949] [1110, 1958] [1110, 1958] [1110, 1958] [73, 1960] [73, 1960]
a
Symmetric positive definite system. p. 27], [507, 1948, p. 336]. With magnetic drum store. d Using magnetic tape for auxiliary storage. b [127, 1948, c
diagonally dominant tridiagonal systems arising in the solution of the heat equation by finite differences. He concluded that the whole procedure of solving this partial differential equation “is stable against round-off error”. It is surprising that Douglas’ paper is little known, because irrespective of the fact that his analysis can be simplified and clarified using modern techniques, his is one of the first truly positive rounding error results to be published. A major breakthrough in the error analysis of GE came with Wilkinson’s pioneering backward error analysis, in which he proved Theorem 9.5 [1085, 1961], [1088, 1963]. Apart from its simplicity and elegance and the realistic nature of the bounds, the main feature that distinguishes Wilkinson’s analysis from the earlier error analyses of GE is that it bounds the normwise backward error rather than the forward error. Wilkinson had been aware of the properties of the growth factor for partial pivoting long before developing his backward error analysis. In a 1954 paper [1081, 1954] he noted that After m reductions the largest element is at most 2”’ times as large as the largest original coefficient. It is possible to construct sets in which this factor is achieved but in practice an increase seldom takes place; more frequently the coefficients become progressively smaller, particularly if the equations are ill-conditioned. This quote summarizes most of what we know today! Four of the first textbooks to incorporate Wilkinson’s analysis were those of Fox [400, 1964, pp. 161-174], Isaacson and Keller [607, 1966], Wendroff [1074,
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1966], and Forsythe and Moler [396, 1967, Chap. 21]. Fox gives a simplified analysis for fixed-point arithmetic under the assumption that the growth factor is of order 1. Forsythe and Moler give a particularly readable backward error analysis that has been widely quoted. Wilkinson’s 1961 result is essentially the best that can be obtained by a normwise analysis. Subsequent work in error analysis for GE has mainly been concerned with bounding the backward error component wise, as in Theorems 9.3 and 9.4. We note that Wilkinson could have given a componentwise bound for the backward perturbation ∆A, since most of his analysis is at the element level. Chartres and Geuder [200, 1967] analyse the Doolittle version of GE. They derive a backward error result (A + ∆A) = b, with a componentwise bound o n ∆A; although they do not recognize it, their bound can be written in the form |∆A| < cnu Reid [867, 1971] shows that the assumption in Wilkinson’s analysis that partial pivoting or complete pivoting is used is unnecessary. Without making any assumptions on the pivoting strategy, he derives for LU factorization the result LU = A + ∆A, |∆ aij| < 3.01 min(i - 1, j) u maxk Again, this is a componentwise bound. Erisman and Reid [355, 1974] note that for a sparse matrix, the term min(i - 1, j) in Reid’s bound can be replaced by mij , where mij is the number of multiplications required in the calculation of lij (i > j) or u ij (i < j). de Boor and Pinkus [273, 1977] give the result stated in Theorem 9.4. They refer to the original 1972 German edition of [955, 198 0] for a proof of the result and explain several advantages to be gained by working with a componentwise bound for ∆A, one of which is the strong result that ensues for totally nonnegative matrices. A result very similar to Theorem 9.4 is proved by Sautter [895, 1978]. Skeel [919, 1979] carried out a detailed componentwise error analysis of GE with a different flavour to the analysis given in this chapter. His aim was to understand the numerical stability of GE (in a precisely defined sense) and to determine the proper way to scale a system by examining the behaviour of the backward and forward errors under scaling (see §9.7). He later used this analysis to derive important results about fixed precision iterative refinement (see Chapter 11). Skeel’s work popularized the use of component wise backward error analysis and componentwise perturbation theory. The componentwise style of backward error analysis for GE is now well known, as evidenced by its presence in the text books of Conte and de Boor [237, 198 0]. Golub and Van Loan [370, 198 9] (also the 1983 first edition), and Stoer and Bulirsch [955, 1980]. Forward error analyses have also been done for GE. The analyses are more complicated and more difficult to interpret than the backward error analyses. Olver and Wilkinson [810, 1982] derive a posteriori forward error bounds that
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require the computation of A-1. Further results are given in a series of papers by Stummel [965, 1982], [966, 1985], [967, 1985], [968, 1985]. Finally, probabilistic error analysis for GE is given by Barlow and Bareiss [63, 1985].
9.7. Scaling Prior to solving a linear system Ax = b by GE we are at liberty to scale the rows and the columns: (9.20) where D1 and D 2 are nonsingular diagonal matrices. We apply GE to the scaled system A’y = c and then recover x from x = D2 y. Although scaling was used in some of the earliest published programs for GE [396, 1967], [745, 1962], how best to choose a scaling is still not well understood, and no single scaling algorithm can be guaranteed always to perform satisfactorily. Wilkinson’s remark “We cannot decide whet her equations are ill-conditioned without examining the way in which the coefficients were derived” [1089, 1965, p. 198] sums up the problem of scaling rather well. The effect of scaling in GE without pivoting is easy to describe. If the elements of D 1 and D2 are powers of the machine base β (so that the scaling is done without error) and GE produces and satisfying A + ∆A = then GE on A’ = D 1 A D2 produces and satisfying A’ + D 1 ∆AD2 = In other words, the rounding errors in GE scale in the same way as A. This is a result of Bauer [78, 1963] (see [396, 19 6 7 , Chap. 11] for a clear proof and discussion). With partial pivoting, however, the choice of pivots is affected by the row scaling (though not the column scaling), and in a way that is difficult to predict. We can take a method-independent approach to scaling, by considering any method for solving Ax = b that yields a solution satisfying
with cn a constant. For the scaled system (9.20) we have
so it is natural to choose D 1 and D2 to minimize As we saw in §7.3 (Theorem 7.8), the minimum possible value is no larger than p ( |A- 1 ||A|) . However, a column scaling has the (usually) undesirable effect of changing the norm in which the error is measured. With row scaling only, the minimum
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value of is cond(A) = achieved when D 1 A has rows of unit 1-norm (see (7.12)). Thus row equilibration yields a cond-bounded forward error. For GE, though, it is possible to do even better. Skeel [919, -1 1979] shows that for D 1 = diag(|A||x|) , the forward error bound for GEPP is proportional to cond(A, x) = the catch is, of course, that the scaling depends on the unknown solution x! Row equilibration can be regarded as approximating x by e in this “optimal” scaling. The LINPACK LU factorization routines do not include scaling, while in the LAPACK driver routine xGESVX an initial scaling is optional. One reason why scaling is not popular with numerical analysts is that a cond( A , x)bounded forward error and a small componentwise relative backward error are both achieved by fixed precision iterative refinement (assuming it converges): see Chapter 11. Even Skeel’s optimal scaling does not guarantee a small componentwise relative backward error. Some programs for GEPP incorporate row scaling implicitly. They compute row scale factors d 1. . . . , dn . but. instead of applying GEPP to diag(d i )-1 × A, they apply it to A and choose as pivot row at the kth stage a row r for which is maximal. This type of scaling has the sole effect of influencing the choice of pivots. There is little justification for using it, and the best bound for the growth factor is 2n -1 multiplied by a product of terms di1 /d i2 that can be large. There is, however. one situation in which a form of implicit row scaling is beneficial. Consider the pivoting strategy that selects as the kth pivot an element for which (9.21) A result of Peña [825, 1994] shows that if there exists a permutation matrix P such that PA has an LU factorization PA = LU with |PA| = |L||U|, then such a factorization will be produced by the pivoting scheme (9.21). This means that, unlike for partial pivoting, we can use the pivoting scheme (9.21) with impunity on totally nonnegative matrices and their inverses, row permutations of such matrices, and any matrix for which some row permutation has the “|PA| = |L||U|” property. However, this pivoting strategy is as expensive as complete pivoting to implement, and for general A it is not guaranteed to produce a factorization as stable as that produced by partial pivoting.
9.8. A Posteriori Stability Tests Having solved a linear system by LU factorization we can compute the componentwise or normwise backward error at the cost of evaluating one or two matrix-vector products (see Theorems 7.1 and 7.3). In some situations,
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though, we may wish to assess the stability of a computed LU factorization before using it to solve one or more linear systems. One possibility is to compute the growth factor by monitoring the size of elements during the elimination, at a cost of O(n 3) comparisons. This has been regarded as rather expensive, and more efficient ways to estimate pn have been sought. Businger [169, 1971] describes a way to obtain an upper bound for p n in O(n2) operations. This approach is generalized by Erisman and Reid [355, 1974], who apply the Holder inequality to the equation
to obtain the bound
(9.22) where p -1 + q-1 = 1. In practice. p = 1,2, are the values of interest. Barlow [56, 1986] notes that application of the Holder inequality instead to
yields a sometimes sharper bound. It is interesting to note that in light of experience with the bound (9.22), Reid [868, 198 7] recommends computing the growth factor explicitly in the context of sparse matrices, arguing that the expense is justified because (9.22) can be a very weak bound. See Erisman et al. [354, 1987] for some empirical results on the quality of the bound. Chu and George [209, 1985] observe that the -norm of the matrix can be computed in O(n 2) operations without forming the matrix explicitly, since Thus one can cheaply compute a bound on from the componentwise backward error bounds in (9.6) and (9.7). All the methods discussed in this section make use of an a priori error analysis to compute bounds on the backward error. Because the bounds do not take into account the statistical distribution of rounding errors, and because they involve somewhat pessimistic constant terms, they cannot be expected to be very sharp. Thus it is important not to forget that it is straightforward to compute the backward error itself: A Exact computation costs a prohibitively expensive O(n 3) operations, but can be estimated in
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O(n 2) operations using the matrix norm estimator in Algorithm 14.4. Another possibility is to use a running error analysis, in which an error bound is computed concurrently with the factors (see 53.3).
9.9. Sensitivity of the LU Factorization Although Theorem 9.3 bounds the backward error of the computed LU factors L and U, it does not give any indication about the size of the forward errors L and U For most applications of the LU factorization it is the backward error and not the forward errors that matters, but it is still of some interest to know how big the forward errors can be. This is a quest ion of perturbation theory and is answered by the next result. Theorem 9.14 (Barrlund, Sun). Let the nonsingular matrices and A+∆A have LU factorizations A = LU and A+∆ A = (L+∆L)(U+∆ U), and assume that ||G||2 < 1, where G = L- 1 ∆AU -1. Then
(9.23) Moreover, if
< 1, where
= (L + ∆L)- 1 ∆A(U + ∆U)-1, then
where stril(·) and triu(·) denote, respectively, the strictly lower triangular part and the upper triangular part of their matrix arguments. The normwise bounds (9.23) imply that x(A) := ||L - 1 ||2 ||U- 1 ||2 ||A|| 2 is an upper bound for the condition numbers of L and U under normwise perturbations. We have
and the ratio x(A)/κ 2 (A) can be arbitrarily large (though if partial pivoting is used then κ2 (L) < n2n - 1 ). The componentwise bounds in Theorem 9.14 are a little unsatisfactory in that they involve the unknown mat rices ∆L and ∆U, but we can set these terms to zero and obtain a bound correct to first order.
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9.10. Notes and References A variant of GE was used by the Chinese around the first century AD; the Jiu Zhang Suanshu (Nine Chapters of the Mathematical Art) contains a worked example for a system of five equations in five unknowns [619, 1991, pp. 156-177]. [696, 1989] Gauss, who was a great statistician and numerical analyst, developed his elimination method as a tool to help him prove results in linear regression theory. The first published appearance of GE is in his Theoria Motus (1809). Stewart [952, 1994] gives a survey of Gauss’s work on solving linear systems: see also the afterword in [423, 1995]. The traditional form of GE, as given at the start of this chapter. can be expressed algorithmically as for k = 1:n for j = k+ 1 :n for i = k + 1:n a i j = a i j - (a i k /a k k )a k j end end end This is identified as the kji form of GE. Altogether there are six possible orderings of the three loops. Doolittle’s method (Algorithm 9.2) is the ijk or jik variant of Gaussian elimination. The choice of loop ordering does not affect the stability of the computation, but can greatly affect the efficiency of GE on a high performance computer. For more on the different loop orderings of GE see Chapter 12; Dongarra, Gustavson. and Karp [310, 1984]; and the books by Dongarra, Duff, Sorensen, and van der Vorst [315, 1991] and Golub and Van Loan [470, 1989]. This chapter draws on the survey paper Higham [545, 1990]. Theorems 9.6 and 9.7 are from Higham and Higham [562, 1989]. Myrick Hascall Doolittle (1830-1913) was a “computer of the United States coast and geodetic survey” [362, 1987]. Crout’s method was published in an engineering journal in 1941 [255, 1941]. GE and its variants were known by various descriptive names in the early days of computing. These include the bordering met hod, the escalator met hod (for matrix inversion), the square root method (Cholesky factorization), and pivotal condensation. A good source for details of these methods is Faddeeva [360, 1959]. In a confidential 1948 report that “covers the general principles of both the design of the [Automatic Computing Engine] and the method of programming adopted for it”, Wilkinson gives a program implementing GE with partial pivoting and iterative refinement [1080, 1948, p. 111]. This was probably the
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first such program to be written and for a machine that had not yet been built! The terms “partial pivoting“ and “complete pivoting” were introduced by Wilkinson in [1085, 1961]. The pivoting techniques themselves were in use in the 1940s and it is not clear who, if anyone, can be said to have invented them: indeed, von Neumann and Goldstine [1057, 1947, §4.2] refer to complete pivoting as the “customary procedure”. There is a long history of published programs for GE. beginning with Crout routines of Forsythe [390, 1960], Thacher [999, 1961], McKeeman [745, 1962], and Bowdler, Martin, Peters, and Wilkinson [138, 1966], all written in Algol 60 (which was the “official” language for publishing mathematical software in the 1960s. and a strong competitor to Fort ran for practical use at that time). The GE routines in LAPACK are the latest in a lineage beginning with the Fortran routines decomp and solve in Forsythe and Moler‘s book [396, 1967], and continuing with routines by Moler [766, 1972], [767, 1972] (which achieve improved efficiency in Fortran by accessing arrays by column rather than by row), Forsythe, Malcolm, and Moler [395, 1977] (these routines incorporate condition estimation-see Chapter 14), and LINPACK [307, 1979]. LU factorization of totally nonnegative matrices has been investigated by Cryer [257, 1973], [258, 1976], Ando [21, 198 7], and de Boor and Pinkus [273, 1977]. It is natural to ask whether we can test for total nonnegativity without computing all the minors. The answer is yes: for an n × n matrix total nonnegativity can be tested in O(n 3) operations. as shown by Gasca and Peña [421, 1992]. The test involves carrying out a modified form of GE in which all the elimination operations are between adjacent rows and then checking whether certain pivots are positive. Note the analogy with positive definiteness, which holds for a symmetric matrix if and only if all the pivots in GE are positive. The dilemma of whether to define the growth factor in terms of exact or computed quantities is faced by all authors; most make one choice or the other, and go on to derive, without comment, bounds that are strictly incorrect. Theorem 9.8, for example, bounds the exact growth factor; the computed one could. conceivably violate the bound, but only by a tiny relative amount. van Veldhuizen [1045, 1977] shows that for a variation of partial pivoting that allows either a row or column interchange at each stage, the growth factor defined in terms of computed quantities is at most about (1 + 3 nu)2n - 1 , compared with the bound 2n -1 for the exact growth factor. The idea of deriving error bounds for GE by analysing the equations obtained by solving A = LU is exploited by Wilkinson [1097, 1974], who gives a general analysis that includes Cholesky factorization. This paper gives a concise summary of error analysis of factorization methods for linear equations and least squares problems. Various authors have tabulated growth factors in extensive tests with ran-
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dom matrices. In tests during the development of LINPACK, the largest value observed was = 23, occurring for a random matrix of 1s, 0s, and -1s [307, 1979, p. 1.21]. Macleod [720, 1989] recorded a value = 35.1, which occurred for a symmetric matrix with elements from the uniform distribution on [-1, 1]. In one MATLAB test of 1000 matrices of dimension 100 from the normal N(0, 1) distribution, I found the largest growth factor to be = 9.59. Gould [474, 1991] used the optimization LANCELOT [236, 1992] to maximize the nth pivot for complete pivoting as a function of about n 3/3 variables comprising the intermediate elements of the elimination; constraints were included that normalize the matrix A, describe the elimination equations, and impose the complete pivoting conditions. Gould’s package found many local maxima, and many different starting values had to be tried in order to locate the matrix for which > 13. In an earlier attempt at maximizing the growth factor, Day and Peterson [271, 1988] used a problem formulation in which the variables are the n2 elements of A, which makes the constraints and objective function substantially more nonlinear than in Gould’s formulation. Using the package NPSOL [444, 1986], they obtained “largest known” growth factors for 5 < n < 7. Theoretical progress into understanding the behaviour of the growth factor for complete pivoting has been made by Day and Peterson [271, 1988], Puschmann and Cortés [849, 1983], Puschmann and Nordio [850, 1985], and Edelman and Mascarenhas [345, 1995]. A novel alternative to partial pivoting for stabilizing GE is proposed by Stewart [942, 1974]. The idea is to modify the pivot element to make it suitably large, and undo this rank one change later using the Sherman-Morrison formula. Stewart gives error analysis that bounds the backward error for this modified form of GE. Theorem 9.8 is proved for matrices diagonally dominant by columns by Wilkinson [1085, 1961, pp. 288-289]. Theorem 9.9 is proved in the same paper. That pn < 2 for matrices diagonally dominant by rows does not appear to be well known, but it is proved by Wendroff [1074, 1966, pp. 122-123]. for example. The results in §9.5 for tridiagonal matrices are taken from Higham [541, 1990 ]. Another method for solving tridiagonal systems is cyclic reduction, which was developed in the 1960s [171, 1970]. Error analysis given by Amodio and Mazzia [15, 1994] shows that cyclic reduction is normwise backward stable for diagonally dominant tridiagonal matrices. The chapter “Scaling Equations and Unknowns” of Forsythe and Moler [396, 1967] is a perceptive, easy to understand treatment that is still well worth reading. Early efforts at matrix scaling for GE were directed to equilibrating either just the rows or the rows and columns simultaneously (so that all the rows and columns have similar norms). An algorithm with the latter aim is described by Curtis and Reid [259, 1972]. Other important references on
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scaling are the papers by van der Sluis [1040, 1970] and Stewart [945, 1977]. which employ normwise analysis, and those by Skeel [919, 1979], [921, 1981], which use componentwise analysis. Much is known about the existence and stability of LU factorizations of M-matrices and related matrices. A is an H-matrix if the comparison matrix M(A) (defined in (8.6)) is an M-matrix. Funderlic, Neumann, and Plemmons [410, 1982] prove the existence of an LU factorization for an H-matrix A that is generalized diagonally dominant, that is, DA is diagonally dominant by columns for some nonsingular diagonal matrix D: they show that the growth factor satisfies pn < 2 maxi |dii |/min i |dii |. Neumann and Plemmons [791, 1984] obtain a growth factor bound for an inverse of an H -matrix. Ahac, Buoni, and Olesky [7, 1988] describe a novel column-pivoting scheme for which the growth factor can be bounded by n then A is an H-matrix. The normwise bounds in Theorem 9.14 are due to Barrlund [71, 1991] and the componentwise ones to Sun [972, 1992]. Similar bounds are given by Stewart [951, 1993] and Sun [973, 1992]. Barrlund [72, 1992] describes a general technique for deriving matrix perturbation bounds using integrals. Interval arithmetic techniques (see §24.4) are worth considering if high accuracy or guaranteed accuracy is required when solving a linear system. We mention just one paper, that by Demmel and Krückeberg [297, 1985], which provides a very readable introduction to the subject and contains further references. For several years Edelman has been collecting information on the solution of large, dense linear algebra problems. His papers [337, 1991], [341, 1993], [342, 1994] present statistics and details of the applications in which large dense problems arise. Edelman also discusses relevant issues such as what users expect of the computed solutions and how best to make use of parallel computers. Table 9.2 contains “world records” for linear systems from Edelman’s surveys. For all the records shown the matrix was complex and the system was solved in double precision arithmetic by some version of LU factorization. Most of the very large systems currently being solved come from the solution of boundary integral equations, a major application being the analysis of radar cross sections; the resulting systems have coefficient mat rices that are complex symmetric (but not Hermitian). A recent reference is Wang [1064, 1991].
9.10.1. LAPACK Driver routines xGESV (simple) and xGESVX (expert) use LU factorization with partial pivoting to solve a general system of linear equations with multiple right-hand sides. The expert driver incorporates iterative refinement, condition estimation, and backward and forward error estimation and has an option to scale the system AX = B to = before solution,
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Table 9.2. Records for largest dense linear systems solved (dimension n). Year 1991 1992/3 1994 1995
n 55,296 75,264 76,800 128,600
Computer Connection Machine CM-2 Intel iPSC/860 Connection Machine CM-5 Intel Paragon
Time 4.4 days 22/3days 4.1 days 1 hour
where D R = diag(ri ) = diag(maxj |aij|) and DC = diag(maxi ri |aij|); the scaling is done by the routine xGEEQU. The LU factorization is computed by the routine xGETRF, which uses a partitioned outer product algorithm. The expert driver also returns the quantity ||A||/||U||, where ||A|| := maxi,j |aij|, which is an estimate of the reciprocal of the growth factor, A value much less than 1 signals that the condition estimate and forward error bound could be unreliable. For band matrices, the driver routines are xGBSV and xGBSVX, and for tridiagonal matrices, xGTSV and xGTSVX; again, these use LU factorization with partial pivoting.
Problems 9.1. (Completion of proof of Theorem 9.1.) Show that if a singular matrix has a unique LU factorization then Ak is nonsingular for k = 1 :n - 1. 9.2. Define A(σ) = A - σI, where and For how many values of σ, at most, does A(σ) fail to have an LU factorization without pivoting? 9.3. Show that to the field of values
has a unique LU factorization if 0 does not belong
9.4. State analogues of Theorems 9.3 and 9.4 for LU factorization with row and column interchanges: PAQ = LU. 9.5. Give a 2 × 2 matrix A having an LU factorization A = LU such that |L||U| < c|A| does not hold for any c, yet is of order 1. 9.6. Show that if is nonsingular and totally nonnegative it has an LU factorization A = LU with L > 0 and U > 0. (Hint: use the inequality
which holds for any totally nonnegative A [414, 1959, p. 100].) Deduce that the growth factor for GE without pivoting p n 1.
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9.7. Show that if is nonsingular and its inverse is totally nonnegative then it has an LU factorization A = LU with |A| = |L||U| . (Use the fact that if C is totally nonnegative and nonsingular then JC - 1 J is totally nonnegative, where J = diag((-1) i +1) (this can be proved using determinantal identities; see [21, 1987, Thm. 3.3]).) 9.8. Show that Theorem 9.5 is valid for GE without pivoting, with a different constant. 9.9. Suppose that GE without pivoting is applied to a linear system Ax = b , where is nonsingular, and that all operations are performed exactly except for the division determining a single multiplier lij (where i > j and A = LU ), which is computed with relative error l i j = l i j(1 + ). Evaluate the difference x between the exact and computed solutions. (The answer allows us to see clearly the effect of a computational blunder, which could, for example. be the result of the malfunction of a computer’s divide operation.) 9.10. Show that θ in Theorem 9.7 satisfies
Hence, for g(n) defined in (9.14) and Sn in (9.11). deduce a larger lower bound than g (2n) > 9.11. Explain the errors in the following criticism of GE with complete pivoting. Gaussian elimination with complete pivoting maximizes the pivot at each stage of the elimination. Since the product of the pivots is the determinant (up to sign), which is fixed, making early pivots large forces later ones to be small. These small pivots will have large relative errors due to the accumulation of rounding errors during the algorithm, and dividing by them therefore introduces larger errors. 9.12. In sparse matrix computations the choice of pivot in GE has to be made with the aim of preserving sparsity as well as maintaining stability. In threshold pivoting, a pivot element is chosen from among those elements in column k that satisfy where is a parameter (see, for example, Duff, Erisman, and Reid [325, 1986, §5.4]). Show that for threshold pivoting
where µ j is the number of nonzero entries in the jth column of U. Hence obtain a bound for p n .
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9.13. (RESEARCH PROBLEM) Obtain sharp bounds for the growth factor for GE with partial pivoting applied to (a) a matrix with lower bandwidth p and upper bandwidth q (thus generalizing Theorem 9.10), and (b) a quasitridiagonal matrix (an n × n matrix that is tridiagonal except for nonzero (1, n) and (n, 1) elements). 9.14. (R ESEARCH P ROBLEM ) Explain why the growth factor for GE with partial pivoting is almost always small in practice. 9.15. (R ESEARCH P ROBLEM ) For GE with complete pivoting what is the value of limn g (n)/n (see (9.14))? Is equal to n for Hadamard matrices?
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Chapter 10
Cholesky Factorization
The matrix of that equation system is negative definite-which is a positive definite system that has been multiplied through by – 1. For all practical geometries the common finite difference Laplacian operator gives rise to these, the best of all possible matrices. Just about any standard solution method will succeed, and many theorems are available for your pleasure. —FORMAN S. ACTON, Numerical Methods That Work (1970) Many years ago we made out of half a dozen transformers a simple and rather inaccurate machine for solving simultaneous equations—the solutions being represented as flux in the cores of the transformers. During the course of our experiments we set the machine to solve the equations— X + Y + Z = 1 X + Y + Z = 2 X + Y + Z = 3 The machine reacted sharply—it blew the main fuse and put all the lights out. —B. V. BOWDEN, The Organization of a Typical Machine (1953) There does seem to be some misunderstanding about the purpose of an a priori backward error analysis. All too often, too much attention is paid to the precise error bound that has been established. The main purpose of such an analysis is either to establish the essential numerical stability of an algorithm or to show why it is unstable and in doing so to expose what sort of change is necessary to make it stable. The precise error bound is not of great importance. —J. H. WILKINSON, Numerics/ Linear Algebra on Digits/ Computers (1974) 203
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10.1. Symmetric Positive Definite Symmetric positive definiteness is one of the highest accolades to which a matrix can aspire. Symmetry confers major advantages and simplifications in the eigenproblem and, as we will see in this chapter, positive definiteness permits economy and numerical stability in the solution of linear systems. A symmetrical matrix is positive definite if xTAx > 0 for all nonzero Well-known equivalent conditions to A = AT being positive definite are • det(A k) > 0, k = 1:n, where Ak = A(1:k, 1:k) is the leading principal submatrix of order k. •
>0, k = 1:n, where
denotes the kth largest eigenvalue.
The first of these conditions implies that A has an LU factorization, A = LU (see Theorem 9.1). Another characterization of positive definiteness is that the pivots in LU factorization are positive, since u kk = det(A k)/det(A k–1). By factoring out the diagonal of U and taking its square root, the LU factorization can be converted into a Cholesky factorization: A = RTR, where R is upper triangular with positive diagonal elements. This factorization is so important that it merits a direct proof. Theorem 10.1. If unique upper triangular A = R T R.
is symmetric positive definite then there is a with positive diagonal elements such that
Proof. The proof is by induction. The result is clearly true for n = 1. Assume it is true for n – 1. The leading principal submatrix An –1 = A(1:n–1,1:n–1) is positive definite, so it has a unique Cholesky factorization A n–1 = We have a factorization
(10.1) if (10.2) (10.3) Equation (10.2) has a unique solution since Rn-1 is nonsingular. Then (10.3) gives β 2 = a – rTr. It remains to check that there is a unique real, positive β satisfying this equation. From the equation 0 < det(A) = det(R T) det(R) = det(R n- 1 ) 2 β 2
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we see that β 2 >0, hence there is a unique β > 0. The proof of the theorem is constructive, and provides a way to compute the Cholesky factorization that builds R a column at a time. Alternatively, we can work directly from the equations
which follow by equating (i, j) elements in A = RTR. By solving these equations in the order (1,1), (1,2), (2,2), (1,3), (2,3), (3,3), . . . , (n,n), we obtain the following algorithm. Algorithm 10.2. Given a symmetric positive definite rithm computes the Cholesky factorization A = RTR.
this algo
for j = 1:n for i = 1:j–1 end end Cost: n 3/3 flops (half the cost of LU factorization). As for Gaussian elimination (GE), there are different algorithmic forms of Cholesky factorization. Algorithm 10.2 is the jik or “sdot” form. We describe the kij, outer product form in §10.3. Given the Cholesky factorization A = RTR, a linear system Ax = b can be solved via the two triangular systems RTy = b and Rx = y. If we define D = then the Cholesky factorization A = R T R T can be rewritten as A = LDL , where L = RT diag(r i i )–1 is unit lower triangular. The LDLT factorization is sometimes preferred over the Cholesky factorization because it avoids the need to compute the n square roots that determine the rii . The LDLT factorization is certainly preferred for solving tridiagonal systems, as it requires n less divisions than Cholesky factorization in the substitution stage. All the results for Cholesky factorization in this chapter have analogues for the LDLT factorization. Block LDLT factorization for indefinite matrices is discussed in §10.4.
10.1.1. Error Analysis Error bounds for Cholesky factorization are derived in a similar way to those for LU factorization. Consider Algorithm 10.2. Using Lemma 8.4 we have (10.4)
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From a variation of Lemma 8.4 in which the division is replaced by a square root (see Problem 10.3), we have
A backward error result is immediate. Theorem 10.3. If Cholesky factorization applied to the symmetric positive runs to completion then the computed factor R definite matrix satisfies (10.5) Theorem 10.4. Let be symmetric positive definite and suppose Cholesky factorization produces a computed factor R and a computed solution to Ax = b. Then (10.6) Proof. The proof is analogous to the proof of Theorem 9.4. These results imply that Cholesky factorization enjoys perfect normwise backward stability. The key inequality is
whose analogue for the computed
is, from (10.5),
Thus (10.6) implies (10.7) where for the last inequality we have assumed that nγ n +1 < 1/2. Another indicator of stability is that the growth factor for GE is exactly 1 (see Problem 10.4). It is important to realize that the multipliers can be arbitrarily large (consider, for example, as θ 0). But, remarkably, for a positive definite matrix the size of the multipliers has no effect on stability. Note that the perturbation ∆A in (10.6) is not symmetric, in general, because the backward error matrices for the triangular solves with R and RT are not the transposes of each other. For conditions guaranteeing that a “small” symmetric ∆A can be found, see Problem 7.11. The following rewritten version of Theorem 10.3 provides further insight into Cholesky factorization.
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Theorem 10.5 (Demmel). If Cholesky factorization applied to the symmetric positive definite matrix runs to completion then the computed factor satisfies
where di = Proof. Theorem 10.3 shows that = A+∆A with |∆ A| < γn + 1 Denoting by the ith column of we have
so that
Then, using the Cauchy–Schwarz inequality,
giving (10.8) and the required bound for ∆A. Standard perturbation theory applied to Theorem 10.4 yields a bound of the form However, with the aid of Theorem 10.5 we can obtain a potentially much smaller bound. The idea is to write A = DHD where D = diag(A )1/2, so that H has unit diagonal. van der Sluis’s result (Corollary 7.6) shows that (10.9) so D is nearly a condition-minimizing diagonal scaling. It follows that κ 2 (H) < nκ 2 (A) and that κ 2 (H) nγn +1/(1– γn +1) then Cholesky factorization applied to A succeeds (barring underflow and overflow) and produces a nonsingular I f min(H) < –nγn +1/(1– γ n+1) then the computation is certain to fail.
10.2 S ENSITIVITY
OF THE
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Proof. The proof is by induction. Consider Algorithm 10.2. The first stage obviously succeeds and gives > 0, since a 11 > 0. Suppose the algorithm has successfully completed k – 1 stages, producing a nonsingular R k-1, and consider equations (10.1)-(10.3) with n replaced by k. The kth stage can be completed, but may give a pure imaginary (it will if fl(a – < 0). However, in the latter event, the error analysis of Theorem 10.5 is still valid! Thus we obtain satisfying = Ak + ∆A k, |∆Ak| < (1 – where
Now, with D k = diag(d k),
we have
using the interlacing of the eigenvalues [470, 1989, Cor. 8.1.4] and the condition of the theorem. Hence is positive definite, and therefore so is the congruent matrix Ak + ∆A k, showing that must be real and nonsingular, as required to complete the induction. If Cholesky succeeds, then, by Theorem 10.5, D –1 (A + ∆A)D –1 is positive definite and so 0 < Hence if min(H) < -nγ n+1/(1– γ n +1) then the computation must fail. Note that, since ||H||2 > 1, the condition for success of Cholesky factorization can be written as κ 2 (H)nγn +1/(1– γ n + 1 ) < 1 .
10.2. Sensitivity of the Cholesky Factorization The Cholesky factorization has perturbation bounds that are similar to those for LU factorization, but of a simpler form thanks to the positive definiteness (||A- 1 ||2 replaces ||U- 1||2 ||L - 1 ||2 in the normwise bounds). Theorem 10.8 (Sun). Let be symmetric positive definite with the Cholesky factorization A = RTR and let AA be a symmetric matrix satisfying ||A- 1 ∆A|| 2 < 1. Then A + ∆A has the Cholesky factorization A + ∆A = (R + ∆R)T(R + ∆R), where
210 Moreover, if
C HOLESKY < 1, where
F ACTORIZATION
= ( R + ∆R) - T ∆A(R + ∆R)-1, then
| ∆R| < triu where triu(·) denotes the upper triangular part. Note that the Cholesky factor of Ak = A(1:k,1:k) is Rk, and κ 2 (A k+l) > κ2 (A k) by the interlacing property of the eigenvalues. Hence if Ak +1 (and hence A) is ill conditioned but Ak is well conditioned then Rk will be relatively insensitive to perturbations in A but the remaining columns of R will be much more sensitive.
10.3. Positive Semidefinite Matrices If A is symmetric and positive semidefinite (xTAx > 0 for all x) then a Cholesky factorization exists, but the theory and computation are more subtle than for positive definite A. The questions of existence and uniqueness of a Cholesky factorization are answered by the following result. Theorem 10.9. Let be positive semidefinite of rank r. (a) There exists at least one upper triangular R with nonnegative diagonal elements such that A = RTR. (b) There is a permutation Π such that Π T AP has a unique Cholesky factorization, which takes the form (10.11) where R11 is r × r upper triangular with positive diagonal elements. Proof. (a): Let the symmetric positive semidefinite square root X of A have the QR factorization X = QR with rii > 0. Then A = X 2 = XTX = R T Q T QR = R T R. (b): The algorithm with pivoting described below amounts to a constructive proof. Note that the factorization in part (a) is not in general unique. For example,
For practical computations a factorization of the form (10.11) is needed, because this factorization so conveniently displays the rank deficiency. Such a factorization can be computed using an outer product Cholesky algorithm, comprising r = rank(A) stages. At the kth stage, a rank-1 matrix is subtracted from A so as to introduce zeros into positions k:n in the k th row and
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column. Ignoring pivoting for the moment, at the start of the kth stage we have
(10.12) where = [0, . . . , 0, rii , . . ., rin ]. The reduction is carried one stage further by computing
Overall we have,
To avoid breakdown when vanishes (or is negative because of rounding errors), pivoting is incorporated into the algorithm as follows. At the start of the kth stage an element > 0 (s > k) is selected as pivot, and rows and columns k and s of Ak, and the kth and sth elements of ri , i = 1:k – 1, are interchanged. The overall effect is to compute the decomposition (10.11), where the permutation Π takes account of all the interchanges. The standard form of pivoting is defined by
This is equivalent to complete pivoting in GE, since Ak is positive semidefinite so its largest element lies on the diagonal. We note for later reference that this pivoting strategy produces a matrix R that satisfies (cf. Problem 18.5) (10.13) It will be convenient to denote by cp(A) := ΠT AΠ the permuted matrix obtained from the Cholesky algorithm with complete pivoting.
10.3.1. Perturbation Theory In this section we analyse, for a positive semidefinite matrix, the effect on the Cholesky factorization of perturbations in the matrix. This perturbation theory will be used in the error analysis of the next section.
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Throughout this section A is assumed to be an n × n positive semidefinite matrix of rank r whose leading principal submatrix of order r is positive definite. For k = 1:r we will write
(10.14) and other matrices will be partitioned conformably. We have the identity
(10.15) where R11 is the Cholesky factor of A 11, R12 =
and
is the Schur complement of A11 in A. Note that Sr( A) 0, so that for k = r, (10.15) is the (unique) Cholesky factorization of A. The following lemma shows how S k(A) changes when A is perturbed. Lemma 10.10. Let E be symmetric and assume A 11 + E 11 is nonsingular. Then (10.16) where W = Proof. We can expand
The result is obtained by substituting this expansion into Sk( A+E) = (A 22 + E22) – (A 12 + E1 2 ) T (A11 + E1 1 ) – 1 ( A 12 + E12), and collecting terms. Lemma 10.10 shows that the sensitivity of S k( A ) to perturbations in A is governed by the matrix W = The question arises of whether, for a given A, the potential magnification of E indicated by (10.16) is attainable. For the no-pivoting strategy, Π = I, the answer is trivially “yes”, since we can take E = with |γ| small, to obtain ||Sk (A+E) For complete pivoting, however, the answer is complicated by the possibility that the sequence of pivots will be different for A+ E than for A, in which case Lemma 10.10 is not applicable. Fortunately, a mild assumption on A is enough to rule out this technical difficulty, for small ||E|| 2. In the next lemma we redefine A := cp(A ) in order to simplify the not at ion.
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Lemma 10.11. Let A := cp(A). Suppose that ( S i ( A ) ) 11 > (S i ( A ))j j, j = 2: n - i , i = 0: r- 1
(10.17)
(where S 0 (A) := A). Then, for sufficiently small ||E||2 , A+E = cp(A + E) . For E = with |γ| sufficiently small,
Proof. Note that since A = cp(A ), (10.17) simply states that there are no ties in the pivoting strategy (since (S i ( A ) ) 11 in (10.12)). Lemma 10.10 shows that Si (A+E) = Si (A) + O(||E|| 2), and so, in view of (10.17), for sufficiently small ||E||2 we have
This shows that A + E = c p(A+E). The last part then follows from Lemma 10.10. We now examine the quantity ||W|| 2 = We show first that ||W|| 2 can be bounded in terms of the square root of the condition number of A1 1 . Lemma 10.12. If A, partitioned as in (10.14), is symmetric positive definite and A11 is positive definite then Proof. Write together with from the fact that the Schur complement inite.
which follows is positive semidef-
Note that, by the arithmetic-geometric mean inequality (x, y > 0), we also have, from Lemma 10.12, ||A 2 2 || 2 )/2. The inequality of Lemma 10.12 is attained for the positive semidefinite matrix
where Ip,q is the p × q identity matrix. This example shows that ||W||2 can be arbitrarily large. However, for A := cp(A ), ||W||2 can be bounded solely in terms of n and k. The essence of the proof, in the next lemma, is that large elements in are countered by small elements in A12. Hereafter we set k = r, the value of interest in the following sections.
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L e m m a 1 0 . 1 3 . Let A := cp(A) and set k = r. Then (10.18) There is a parametrized family of rank-r matrices A(θ ) = cp(A(θ )), for which
Proof. The proof is a straightforward computation. The matrix A(O) := R(θ)TR(θ), where
(10.19) with c = cosθ, s = sin θ. This is the r × n version of Kahan’s matrix (8.10). R satisfies the inequalities (10.13) (as equalities) and so A(θ ) = cp(A(θ )). We conclude this section with a “worst-case” example for the Cholesky factorization with complete pivoting. Let U( θ) = diag(r, r–1, . . . . 1)R(θ), where R(θ) is given by (10.19), and define the rank-r matrix C(θ) = U(θ) T U(θ). Then C(θ) satisfies the conditions of Lemma 10.11. Also,
Thus, from Lemma 10.11, for E =
with |γ| and θ sufficiently small,
This example can be interpreted as saying that in exact arithmetic the residual after an r-stage Cholesky factorization of a semidefinite matrix A can overestimate the distance of A from the rank-r semidefinite matrices by a factor as large as (n – r)(4r – 1)/3.
10.3.2. Error Analysis In this section we present a backward error analysis for the Cholesky factorization of a positive semidefinite matrix. An important consideration is that
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a matrix of floating point numbers is very unlikely to be ‘(exactly” positive semidefinite; errors in forming or storing A will almost certainly render the smallest eigenvalue nonzero, and possibly negative. Therefore error analysis for a rank r positive semidefinite matrix may appear, at first sight, to be of limited applicability. One way around this difficulty is to state results for A = A + ∆A, where A is the matrix stored on the computer, A is positive semidefinite of rank r, and ∆A is a perturbation, which could represent the rounding errors in storing Ã, for example. However, if the perturbation AA is no larger than the backward error for the Cholesky factorization, then this extra formalism can be avoided by thinking of ∆A as being included in the backward error matrix. Hence for simplicity, we frame the error analysis for a positive semidefinite A. The analysis makes no assumptions about the pivoting strategy, so A should be thought of as the pre-permuted matrix Π T AΠ. Theorem 10.14. Let A be an n × n symmetric positive semidefinite matrix of floating point numbers of rank r < n. Assume that A11 = A(1:r, 1:r) is positive definite with (10.20) where A11 = D 11 H11 D 11 and D11 = diag(A 11)1/2. Then, in floating point arithmetic, the Cholesky algorithm applied to A successfully completes r stages (barring underflow and overflow), and the computed r × n Cholesky factor satisfies (10.21) where W = Proof. First, note that condition (10.20) guarantees successful completion of the first r stages of the algorithm by Theorem 10.7. Analysis very similar to that leading to Theorem 10.3 shows that (10.22) where
and (10.23)
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Taking norms in (10.23) and using the inequality we obtain
which implies (10.24) Our aim is to obtain an a priori bound for It is clear from ( r +1 ) (10.22)-(10.24) that to do this we have only to bound ||Â ||2. To this end, we interpret (10.22) in such a way that the perturbation theory of §10.3.1 may be applied. Equation (10.22) shows that is the true Schur complement for the matrix A + E and that is positive definite. Hence we can use Lemma 10.10 to deduce that
Substituting from (10.24) we find that
Finally, using (10.22) and (10.24), we obtain
Theorem 10.14 is just about the best result that could have been expected, because the bound (10.21) is essentially the same as the bound obtained on taking norms in Lemma 10.10. In other words, (10.21) simply reflects the inherent mathematical sensitivity of A–RTR to small perturbations in A. We turn now to the issue of stability. Ideally, for A as defined in Theorem 10.14, the computed Cholesky factor Rr produced after r stages of the algorithm would satisfy
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where cn is a modest constant. Theorem 10.14 shows that stability depends on the size of ||W||2 = (to the extent that ||W||2 appears in a realistic bound for the backward error). If no form of pivoting is used then ||W|| 2 can be arbitrarily large for fixed n (see §10.3.1) and the Cholesky algorithm must in this case be classed as unstable. But for complete pivoting we have from Lemma 10.13 the upper bound ||W|| 2 < (1/3(n – r)(4 r – 1))1/2. Thus the Cholesky algorithm with complete pivoting is stable if r is small, but stability cannot be guaranteed, and seems unlikely in practice, if ||W|| 2 (and hence, necessarily, r and n) is large. Numerical experiments show that ||W||2 is almost always small in practice (typically less than 10) [540, 1990]. However, it is easy to construct examples where ||W||2 is large. For example, if R is a Cholesky factor of A from complete pivoting then let C = M(R ) T M( R), where M( R) is the comparison matrix; C will usually have a much larger value of ||W|| 2 than A. An important practical issue is when to terminate the Cholesky factorization of a semidefinite matrix. The LINPACK routine xCHDC proceeds with the factorization until a nonpositive pivot is encountered, that is, up to and including stage k – 1, where k is the smallest integer for which (10.25) Usually k > r + 1, due to the effect of rounding errors. A more sophisticated termination criterion is to stop as soon as (10.26) for some readily computed norm and a suitable tolerance This criterion terminates as soon as a stable factorization is achieved, avoiding unnecessary work in eliminating negligible elements in the computed Schur complement Note that is indeed a reliable order-of-magnitude estimate of the true residual, since is the only nonzero block of Â(k) and, by (10.22) and with ||E|| = O(u)(||A|| + ||Â(k)||). (10.24), Another possible stopping criterion is (10.27) This is related to (10.26) in that if A (pre-permuted) and Âk are positive semidefinite then = maxi,j |aij| ||A|| 2, and similarly max Note that (10.27) bounds since if (10.27) holds first at the kth stage then, using Theorem 8.13,
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Practical experience shows that the criteria (10.26) and (10.27) with = nu both work well, and much more effectively than (10.25) [540, 1990]. We favour (10.27) because of its negligible cost.
10.4. Symmetric Indefinite Matrices and Diagonal Pivoting Method Let be symmetric but indefinite, that is, (x T A x) (y T Ay) < 0 for some x and y. How can we solve Ax = b efficiently? Gaussian elimination with partial pivoting (GEPP) can be used to compute the factorization PA = LU, but it does not take advantage of the symmetry to reduce the cost and storage. We might try to construct a factorization A = L D L T, where L is unit lower triangular and D is diagonal. But this factorization may not exist, even if symmetric pivoting is allowed, and if it does exist its computation may be unstable. For example, consider
There is arbitrarily large element growth for 0 < aµ 0 Set s = 1, and choose Π so that |e 11| = µ 1 . else Set s = 2, and choose Π so that |e 21| = µ 0 . end
(0, 1).
Note that µ 1 is the best 1 × 1 pivot under symmetric permutations and µ 0 is the pivot that would be chosen by GE with complete pivoting. This strategy says “as long as there is a diagonal pivot element not much smaller than the complete pivot, choose it as a 1 × 1 pivot”, that is, “choose a 1 × 1 pivot whenever possible”. If the strategy dictates the use of a 2 × 2 pivot then that pivot E is indefinite (see Problem 10.11). It remains to determine a. This is done by minimizing a bound on the element growth. For the following analysis we assume that the interchanges
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have already been done. If s = 1 then
Now consider the case s = 2. The (i, j) element of the Schur complement à = B – CE - 1 CT is (10.28) Now
and, using the symmetry of E,
Since a
(0, 1), we have |det(E)| > (1 – a 2 )
Thus
Since |cij| < µ 0, we obtain from (10.28)
To determine a we equate the maximum growth for two s = 1 steps with that for one s = 2 step:
which reduces to the quadratic equation 4a 2 – a – 1 = 0. We require the positive root
The analysis guarantees a growth factor bound of (1+a- 1 ) n -1 = (2.57) n - 1 . This bound is pessimistic, however; a much more detailed analysis by Bunch [158, 1971] shows that the growth factor is no more than 3.07( n –1)0.446 times larger than the bound (9.13) for LU factorization with complete pivoting—a very satisfactory result. Strictly speaking, bounding the growth factor bounds only ||D||, not ||L||. But it is easy to show that for s = 1 and 2 no element of CE-1 exceeds max{1/a,1/(1–a)} in absolute value, and so ||L|| is bounded independently of A. Since complete pivoting requires the whole active submatrix to be searched at each stage, it needs up to n3/6 comparisons, and so the method is rather expensive.
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10.4.2. Partial Pivoting Bunch and Kaufman [164, 1977] devised a pivoting strategy for the diagonal pivoting method that requires only O( n 2) comparisons. At each stage it searches at most two columns and so is analogous to partial pivoting for LU factorization. The strategy contains several logical tests. As before, we describe the pivoting for the first stage only. Recall that s denotes the size of the pivot block. Choose a
(0, 1).
If = 0 there is nothing to do on this stage of the elimination. r := min{i > 2: |ai1| = if |a 11| > (1) s = 1, Π = I else
if |a1 1 |σ > a 2 (2) s = 1, Π = I else if |arr| > aσ (3) s = 1 and choose Π to swap rows and columns 1 and r. else (4) s = 2 and choose Π to swap rows and columns 2 and r, so that |(ΠAΠ T )21| = end end To understand the algorithm it helps to consider the matrix
and to note that the pivot is one of a 11, arr , and = |ar1|, this matrix with replaced by ar1 ).
(or, rather, since
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To bound the element growth reconsider each case in turn, noting that for cases (1) and (2) the elements of the Schur complement are given by14
Case (l):
Case (2): Using symmetry,
Case (3): The original arr is now the pivot, and |arr| > a σ, so
Case (4): This is where we use a 2 x 2 pivot, which, after the interchanges, is E = (Π T AΠ)(1:2,1:2) = Now
The elements of the Schur complement à = B-CE- 1 C T a r e g i v e n b y
so
This analysis shows that the bounds for the element growth for s = 1 and s = 2 are the same as the respective bounds for the complete pivoting strategy. Hence, using the same reasoning, we again choose a = The growth factor for the partial pivoting strategy is bounded by (2.57)n - 1 . As for GEPP, large growth factors do not seem to occur in practice. But unlike for GEPP, no example is known for which the bound is attained [164, 1977]; see Problem 10.18. 14
We commit a minor abuse of notation, in that in the rest of this section ãij should really be ãi– 1, j–1 (s = 1) or ãi- 2 ,j –2 (s = 2).
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As noted in the previous subsection, a bound on the growth factor p n does not in itself ensure stability. Indeed although ||D||/||A|| is bounded for partial pivoting, ||L||/||A|| can be arbitrarily large for fixed n; see Problem 10.15. Higham [559, 1995] gives a detailed error analysis of the diagonal pivoting method with an arbitrary pivoting strategy, under the assumption that computed solutions to linear systems involving 2 × 2 pivots have a small componentwise relative backward error. The conclusions are that the computed factors satisfy
and the computed solution to a linear system Ax = b satisfies
where p1 and p 2 are linear polynomials. For the partial pivoting strategy, Higham shows that if linear systems involving 2 × 2 pivots are solved by GEPP or by use of the explicit inverse, then the computed solutions do indeed have a small componentwise relative backward error, and that, moreover, || |L||D||LT| ||M < 36n p n||A||M , where ||A||M = max i,j |aij|. Thus the diagonal pivoting method with partial pivoting is stable if the growth factor is small.
10.5. Nonsymmetric Positive Definite Matrices The notion of positive definiteness can be extended to nonsymmetric matrices. A nonsymmetric matrix is positive definite if xTAx > 0 for all x 0. This is equivalent to the condition that the symmetric part AS of A is positive definite, where A A S + A K w i t h A S = ( A + A T )/2 and A K = ( A – A T )/2. A positive definite matrix clearly has nonsingular leading principal submatrices, and so has an LU factorization, A = LU. It can even be shown that pivots uii are positive. However, there is no guarantee that the factorization is stable without pivoting, as the example shows. The standard error analysis for LU factorization applies (Theorems 9.3 and 9.4), and so the question is whether can be suitably bounded. Golub and Van Loan [469, 1979] show that, for the exact LU factors, (10.29) Let which is just κ 2 (A) when A is symmetric. Mathias [731, 1992] shows that || |L||U| ||F (involving now the computed LU factors) is at most a factor 1 + 30un3 / 2 X(A) times larger than the
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upper bound in (10.29), and that the LU factorization (without pivoting) succeeds if 24n3 / 2 X (A )u < 1. These results show that it is safe not to pivot provided that the symmetric part of A is not too ill conditioned relative to the norm of the skew-symmetric part. If A is symmetric (AK = 0) then we recover the results for symmetric positive definite matrices.
10.6. Notes and References André-Louis Cholesky (1875-1918) was a French military officer involved in geodesy and surveying in Crete and North Africa. In some books his name is misspelled “Choleski”. Details of Cholesky’s life-and a discussion about the pronunciation of his name!-can be found in the electronic mail magazine NA-Digest, volume 90, 1990, issues 7, 8, 10–12, and 24; see, in particular, the biography [22, 1922]. Cholesky’s work was published posthumously on his behalf by Benoit [91, 1924]. The properties of the Cholesky factorization are intimately associated with the properties of the Schur complement, as is apparent from some of the proofs in this chapter. The same is true for GE in general. An excellent survey of the Schur complement, containing historical comments, theory, and applications, is given by Cottle [248, 1974]. For results on the Cholesky factorization in Hilbert space see Power [841, 1986]. A book by George and Liu [438, 1981] is devoted to the many practical issues in the implementation of Cholesky factorization for the solution of sparse symmetric positive definite systems. There is no floating point error analysis of Cholesky factorization in Wilkinson’s books, but he gives a detailed analysis in [1092, 1968], showing that = A+E, with ||E||2 < 2.5n3 / 2 u||A|| 2. It is unfortunate that this paper is in a rather inaccessible proceedings, because it is a model of how to phrase and interpret an error analysis. Meinguet [747, 1983] and Sun [973, 1992] give componentwise backward error bounds similar to those in Theorems 10.3 and 10.4. Kielbasinski [657, 198 7] reworks Wilkinson’s analysis to improve the constant. The fact that κ2 (H) can replace the potentially much larger κ 2 (A) in the forward error bound for the Cholesky method was stated informally and without proof by Wilkinson [1092, 1968, p. 638]. Demmel [283, 1989] made this observation precise and explored its implications; Theorems 10.5, 10.6, and 10.7 are taken from [283, 1989]. The bounds in Theorem 10.8 are from Sun [971, 1991], [972, 1992]. Similar bounds are given by Stewart [944, 1977], [951, 1993], Barrlund [71, 1991], and Sun [973, 1992]. A perturbation bound that can be much smaller than
10.6 N OTES
AND
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the normwise one in Theorem 10.8 is derived and explored by Chang and Paige [198, 1995]. Perturbation results of a different flavour, including one for structured perturbations of the form of ∆A in Theorem 10.5, are given by Drmac, Omladc, and Veselic [321, 1994]. The perturbation and error analysis of §10.3 for semidefinite matrices is from Higham [540, 1990], where in a perturbation result for the QR factorization with column pivoting is also given. For an application in optimization that makes use of Cholesky factorization with complete pivoting and the analysis of §10.3.1 see Forsgren, Gill, and Murray [384, 1995]. Fletcher and Powell [382, 1974] describe several algorithms for updating an LDLT factorization of a symmetric positive definite A when A is modified by a rank-1 matrix. They give detailed componentwise error analysis for some of the methods. An excellent way to test whether a given symmetric matrix A is positive (semi) definite is to attempt to compute a Cholesky factorization. This test is less expensive than computing the eigenvalues and is numerically stable. Indeed, if the answer “yes” is obtained, it is the right answer for a nearby matrix, whereas if the answer is “no” then A must be close to an indefinite matrix. See Higham [535, 1988] for an application of this definiteness test. An algorithm for testing the definiteness of a Toeplitz matrix is developed by Cybenko and Van Loan [260, 1986], as part of a more complicated algorithm. According to Kerr [654, 1990], misconceptions of what is a sufficient condition for a matrix to be positive (semi) definite are rife in the engineering literature (for example, that it suffices to check the definiteness of all 2 × 2 submatrices). See also Problem 10.8. For some results on definiteness tests for Toeplitz matrices, see Makhoul [722, 1991]. A major source of symmetric indefinite linear systems is the least squares problem, because the augmented system is symmetric indefinite; see Chapter 19. Other sources of such systems are interior methods for solving constrained optimization problems (see Forsgren, Gill, and Shinnerl [385, 1996], Turner [1030, 1991], and Wright [1115, 1992]) and linearly constrained optimization problems (see Gill, Murray, Saunders, and Wright [445, 1990], [446, 1991 ]). The idea of using a block LDLT factorization with some form of pivoting for symmetric indefinite matrices was first suggested by Kahan in 1965 [166, 1971]. Bunch and Parlett [166, 1971] developed the complete pivoting strategy and Bunch [158, 1971] proved its stability. Bunch [160, 1974] discusses a rather expensive partial pivoting strategy that requires repeated scalings. Bunch and Kaufman [164, 1977] found the efficient partial pivoting strategy presented here, which is the one now widely used, and Bunch, Kaufman and Parlett [165, 1976] give an Algol code implementing the diagonal pivoting method with this pivoting strategy. Dongarra, Duff, Sorensen, and van der Vorst [315, 1991, §5.4.5] show how to develop a partitioned version of the diagonal pivoting
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method with partial pivoting. Liu [709, 198 7] shows how to incorporate a threshold into the Bunch– Kaufinan partial pivoting strategy for sparse symmetric matrices; see also Duff et al. [326, 1991]. The partial pivoting strategy and variants of it described by Bunch and Kaufman [164, 1977] do not preserve band structure, but the fill-in depends on the number of 2 × 2 pivots, which is bounded by the number of negative eigenvalues (see Problem 10.11). Jones and Patrick [615, 1993], [616, 1994] show how to exploit this fact. The complete and partial pivoting strategies of Bunch et al. use a fixed number of tests to determine each pivot. Another possibility is to prescribe growth bounds corresponding to 1 × 1 and 2 × 2 pivots and to search in some particular order for a pivot satisfying the bound. Fletcher [375, 1976] uses this approach to define a pivoting strategy that usually requires only O(n 2) operations. Duff, Reid, and co-workers apply the same approach to the diagonal pivoting method for sparse matrices, where sparsity considerations also influence the choice of pivot [331, 1979], [326, 1991]; their Fortran codes MA27 [329, 1982] and MA47 [330, 1995] implement the methods. Gill, Murray, Ponceleón, and Saunders [443, 1992] show how for sparse, symmetric indefinite systems the diagonal pivoting factorization can be used to construct a (positive definite) preconditioned for an iterative method. Another method for solving symmetric indefinite systems is Aasen’s method [1, 1971], which employs the factorization PAPT = LTL T , where L is unit lower triangular and T is tridiagonal. It is competitive with the diagonal pivoting method in terms of speed. Barwell and George [77, 1976] compare the performance of Fortran codes for several methods for solving symmetric indefinite systems, including the diagonal pivoting method and Aasen’s method. Dax and Kaniel [270, 1977] propose computing a factorization PAPT = LDL T for symmetric indefinite matrices by an extended form of Gaussian elimination in which extra row operations are used to “build up” a pivot element prior to the elimination operations; here, L is unit lower triangular and D is diagonal. A complete pivoting strategy for determining the permutation P is described in [270, 1977] and partial pivoting strategies in Dax [268, 1982]. Analogues of the factorization for symmetric matrices exist for skewsymmetric matrices; see Bunch [161, 1982]. Bunch [159, 1971] shows how to scale a symmetric matrix so that in every nonzero row and column the largest magnitude of an element is 1.
10.6.1. LAPACK Driver routines xPOSV (simple) and xPOSVX (expert) use the Cholesky factorization to solve a symmetric (or Hermitian) positive definite system of linear equations with multiple right-hand sides. (There are corresponding
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routines for packed storage, in which one triangle of the matrix is stored in a one-dimensional array: PP replaces PO in the names.) The expert driver incorporates iterative refinement, condition estimation, and backward and forward error estimation and has an option to scale the system AX = B t o (D – 1 AD– 1 )DX = D – 1 B, where D = diag Modulo the rounding errors in computing and applying the scaling, the scaling has no effect on the accuracy of the solution prior to iterative refinement, in view of Theorem 10.6. The Cholesky factorization is computed by the routine xPOTRF , which uses a partitioned algorithm that computes R a block row at a time. The drivers xPTSV and xPTSVX for symmetric positive definite tridiagonal matrices use LDLT factorization. LAPACK does not currently contain a routine for Cholesky factorization of a positive semidefinite matrix, but there is such a routine in LINPACK (xCHDC ). Driver routines xSYSV (simple) and xSYSVX (expert) use the block LDLT factorization (computed by the diagonal pivoting method) with partial pivoting to solve a symmetric indefinite system of linear equations with multiple right-hand sides. For Hermitian matrices the corresponding routines are xHESV (simple) and xHESVX (expert). (Variants of these routines for packed storage have names in which SP replaces SY and HP replaces HE.) The expert drivers incorporate iterative refinement, condition estimation, and backward and forward error estimation. The factorization is computed by the routine xSYTRF or xHETRF.
Problems 10.1. Show that if
is symmetric positive definite then
What does this statement imply about maxi,j
|aij|?
10.2. If A is a symmetric positive definite matrix, how would you compute x T A- 1 x? 10.3. Let y = any order. Show that
where
be evaluated in floating point arithmetic in
for all i, and |θk + 1| < γ k +1 .
10.4. Let be symmetric positive definite. Show that the reduced submatrix B of order n—1 at the end of the first stage of GE is also symmetric
228 positive definite. Deduce that 0 < that the growth factor pn = 1.
CHOLESKY FACTORIZATION = akk and hence
10.5. Show that the backward error result (10.6) for the solution of a symmetric positive definite linear system by Cholesky factorization implies
where ||A||M = maxi,j |aij| (which is not a consistent matrix norm—see §6.2). The significance of this result is that the bound for ||∆ A||M/||A||M contains a linear polynomial in n, rather than the quadratic that appears for the 2-norm in (10.7). be positive semidefinite of rank r and suppose it 10.6. Let A = cp(A) has the Cholesky factorization (10.11) with Π = I. Show that Z = [W , –I] T is a basis for the null space of A, where W = 10.7. Prove that (10.13) holds for the Cholesky decomposition with complete pivoting. 10.8. Give an example of a symmetric matrix for which the leading principal submatrices Ak satisfy det(A k) > 0, k = 1:n, but A is not positive semidefinite (recall that det(A k) > 0, k = 1:n, implies that A is positive definite). State a condition on the minors of A that is both necessary and sufficient for positive semidefiniteness. 10.9. Suppose the outer product Cholesky factorization algorithm terminates at the (k+1)st stage (see (10.15)), with a negative pivot in the (k + 1, k + 1) position. Show how to construct a direction of negative curvature for A (a vector p such that pTAp < 0). 10.10. What is wrong with the following argument? A positive semidefinite matrix is the limit of a positive definite one as the smallest eigenvalue tends to zero. Theorem 10.3 shows that Cholesky factorization is stable for a positive definite matrix, and therefore, by continuity, it must be stable for a positive semidefinite matrix, implying that Theorem 10.14 is unnecessarily weak (since ||W|| 2 can be large). 10.11. Consider the diagonal pivoting method applied to a symmetric matrix. Show that with complete pivoting or partial pivoting any 2 × 2 pivot is indefinite. Hence give a formula for the inertia in terms of the block sizes of the block diagonal factor. Show how to avoid overflow in computing the inverse of a 2 × 2 pivot. 10.12. Describe the effect of applying the diagonal pivoting method with partial pivoting to a 2 × 2 symmetric matrix. 10.13. What factorization is computed if the diagonal pivoting method with partial pivoting is applied to a symmetric positive definite matrix?
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10.14. (Sorensen and Van Loan; see [315, 1991, §5.3.2]) Suppose the partial pivoting strategy for the diagonal pivoting method is modified by redefining
(thus “σ n e w = m a x (σo l d ,|a rr | )”). Show that the same growth factor bound holds as before and that for a positive definite matrix no interchanges are done and only 1 × 1 pivots are used. 10.15. Let
where 0 < < 1, and suppose the diagonal pivoting method is applied to A, yielding a factorization PAP T = LDL T. Show that with partial pivoting is unbounded as whereas with complete pivoting is bounded independently of 10.16. Let
be nonsymmetric positive definite. Show that the Schur complement S = is also positive definite. In other words, show that GE preserves positive definiteness. 10.17. A matrix of the form
where and are symmetric positive definite has been called a symmetric quasidefinite matrix by Vanderbei [1047, 1995]. Show that (a) A is nonsingular, (b) for any permutation Π, Π T AΠ has an LU factorization, (c) AS is nonsymmetric positive definite, where S = diag(I, –I). (This last property reduces the question of the stability of an LDLT factorization of A to that of the stability of the LU factorization of a nonsymmetric positive definite matrix, for which see §10.5. This reduction has been pointed out and exploited by Gill, Saunders, and Shinnerl [448, 1996].) 10.18. (RESEARCH PROBLEM) Is the growth factor bound (2.57)n –1 for the diagonal pivoting method with partial pivoting attainable? If not, how big can the growth factor be? Similarly, what is a sharp bound for the complete pivoting growth factor? 10.19. (RESEARCH PROBLEM) Bound the growth factor for Aasen’s method [1, 1971].
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Chapter 11
Iterative Refinement
The ILLIAC’s memory is sufficient to accommodate a system of 39 equations when used with Routine 51. The additional length of Routine 100 restricts to 37 the number of equations that it can handle. With 37 equations the operation time of Routine 100 is about 4 minutes per iteration. —JAMES N. SNYDER, On the improvement of the Solutions to a Set of Simultaneous Linear Equations Using the ILLIAC (1955)
In a short mantissa computing environment the presence of an iterative improvement routine can significantly widen the class of solvable Ax = b problems. — GENE H. GOLUB and CHARLES F. VAN LOAN, Matrix Computations (1989)
Most problems involve inexact input data and obtaining a highly accurate solution to an imprecise problem may not be justified. — J. J. DONGARRA, J. R. BUNCH, C. B. MOLER, and G. W. STEWART, LINPACK Users’ Guide (1979)
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Iterative refinement is an established technique for improving a computed solution to a linear system Ax = b. The process consists of three steps: 1. Compute r = b – A 2. Solve Ad = r. 3. Update y =
+ d.
(Repeat from step 1 if necessary, with
replaced by y).
If there were no rounding errors in the computation of r, d, and y, then y would be the exact solution to the system. The idea behind iterative refinement is that if r and d are computed accurately enough then some improvement in the accuracy of the solution will be obtained. The economics of iterative refinement are favorable for solvers based on a factorization of A, because the factorization used to compute can be reused in the second step of the refinement. Traditionally, iterative refinement is used with Gaussian elimination (GE), and r is computed in extended precision before being rounded to working precision. Iterative refinement for GE was used in the 1940s on desk calculators, but the first thorough analysis of the method was given by Wilkinson in 1963 [1088, 196 3]. The behaviour of iterative refinement for GE is usually summarized as follows: if double precision is used in the computation of r, and A is not too ill conditioned, then the iteration produces a solution correct to working precision and the rate of convergence depends on the condition number of A. In the next section we give a componentwise analysis of iterative refinement that confirms this summary and provides some further insight.
11.1. Convergence of Iterative Refinement Let be nonsingular and let be a computed solution to Ax = b. Define x 1 = and consider the following iterative refinement process: ri = b – Axi (precision ), solve Adi = ri (precision u), xi+1 = xi + di (precision u), i = 1, 2, . . . . For traditional iterative refinement, = u 2. Note that in this chapter subscripts specify members of a vector sequence, not vector elements. We henceforth define ri , di , and xi to be the computed quantities (to avoid a profusion of hats). The only assumption we will make on the solver is that the computed solution to a system Ay = c satisfies (11.1) Thus the solver need not be LU factorization or even a factorization method. The page or so of analysis that follows is straightforward but tedious. The reader is invited to jump straight to (11.4), at least on first reading.
11.1 C ONVERGENCE
OF
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Consider first the computation of ri . There are two stages. First, si = f l( b – Axi ) = b – Axi + ∆si is formed in the (possibly) extended precision (cf. (3.10)), where so that |∆ si | < Second, the residual is rounded to the working precision: ri = fl(s i) = si + fi , where |fi | < u|si |. Hence
By writing xi = x + (xi – x), we obtain the bound (11.2) For the second step we have, by (11.1), (A + ∆A i)di = ri . Now write
where, since θi :=
Hence (11.3)
For the last step,
Using (11.3) we have
Hence
Substituting the bound for |∆ ri | from (11.2) gives
(11.4) Note that
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As long as A is not too ill conditioned and the solver is not too unstable, we have which means that the error contracts until we reach a point at which the gi term becomes significant. The limiting normwise accuracy, that is, the minimum size of Moreover, if for some µ, then we can expect to obtain a componentwise relative error of order µu , that is, mini We concentrate now on the case where the solver uses LU factorization. In the traditional use of iterative refinement, = u 2, and one way to summarize our findings is as follows. Theorem 11.1 (mixed precision iterative refinement). Let iterative refinement be applied to the nonsingular linear system Ax = b, using LU factorization and with residuals computed in double the working precision. Let η = u|| |A - 1||L||U| || where L and U are the computed LU factors of A. Then, provided η is sufficiently less than 1, iterative refinement reduces the error by a factor approximately η at each stage, until This theorem is stronger than the standard results in the literature, which have in place of η. We can have η 1. Therefore solving Ax = b to achieve a small componentwise relative backward error can be regarded as an ill-posed problem when |A||x| has a zero component. The quantity σ(A, x) reflects this ill-posedness because it is large when |A||x| has a relatively small component. For a lucid survey of both fixed and mixed precision iterative refinement and their applications, see Björck [111, 1990]. For particular applications of fixed precision iterative refinement, see Govaerts and Pryce [475, 1990] and Jankowski and Wozniakowski [611, 1985]. By increasing the precision from one refinement iteration to the next it is possible to compute solutions to arbitrarily high accuracy, an idea first suggested by Stewart in an exercise [941, 1973, pp. 206–207]. For algorithms, see Kielbasinski [656, 1981] and Smoktunowicz and Sokolnicka [931, 1984]. There are a number of practical issues to attend to when implementing iterative refinement. Mixed precision iterative refinement cannot be implemented in a portable way when the working precision is already the highest precision supported by a compiler. This is the main reason why iterative refinement is not supported in LINPACK. (The LINPACK manual lists a subroutine that implements mixed precision iterative refinement for single precision data, but it is not part of LINPACK [307, 1979, pp. 1.8–1. 10]. ) For either form of refinement, a copy of the matrix A needs to be kept in order to form the residual, and this necessitates an extra n 2 elements of storage. A convergence test for terminating the refinement is needed. In addition to revealing when convergence has been achieved, it must signal lack of (sufficiently fast) convergence, which may certainly be experienced when A is very ill conditioned. In the LAPACK driver xGESVX, fixed precision iterative refinement is terminated if the componentwise relative backward error w = w|A|,|b| satisfies 1. w < u, 2. w has not decreased by a factor of at least 2 during the current iteration, or 3. five iterations have been performed. These criteria were chosen to be robust in the face of different BLAS implementations and machine arithmetics. In an implementation of mixed precision iterative refinement it is more natural to test for convergence of the sequence with a test such as < u (see, e.g., Forsythe and
PROBLEMS
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Moler [396, 1967, p. 65]). However, if A is so ill conditioned that Theorem 11.1 is not applicable, the sequence could converge to a vector other than the solution. This behaviour is very unlikely, and Kahan [626, 1966] quotes a “prominent figure in the world of error-analysis” as saying “Anyone unlucky enough to encounter this sort of calamity has probably already been run over by a truck.” A by-product of extended precision iterative refinement is an estimate of the condition number. Since the error decreases by a factor approximately η = on each iteration (Theorem 11.1), the relative change made to x on the first iteration should be about η , that is, Now that reliable and inexpensive condition estimators are available (Chapter 14) this rough estimate is less important. An unusual application of iterative refinement is to fault-tolerant computing. Boley et al. [132, 1994] propose solving Ax = b by GEPP or QR factorization, performing one step of fixed precision iterative refinement and then testing whether the a priori residual bound in Theorem 11.4 is satisfied. If the bound is violated then a hardware fault may have occurred and special action is taken.
11.3.1. LAPACK Iterative refinement is carried out by routines whose names end -RFS, and these routines are called by the expert drivers (name ending -SVX). Iterative refinement is available for all the standard matrix types except triangular matrices, for which the original computed solution already has a componentwise relative backward error of order u. As an example, the expert driver xGESVX uses LU factorization with partial pivoting and fixed precision iterative refinement to solve a general system of linear equations with multiple right-hand sides, and the refinement is actually carried out by the routine xGERFS .
Problems 11.1. Show that for σ = maxi |xi |/mini |xi |.
and
where
11.2. Use the analysis of §11.1 to show that, under the conditions of Theorem 11.4, is bounded by a multiple of cond(A, x)u for GEPP after one step of fixed precision iterative refinement. 11.3. Investigate empirically the size of
for L from GEPP.
11.4. (Demmel and Higham [291, 1992]) Suppose GEPP with fixed precision iterative refinement is applied to the multiple-right-hand side system AX = B, and that refinement of the columns of X is done “in parallel”: R = B – AX,
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AD = R, Y = X + D. What can be said about the stability of the process if R is computed by conventional multiplication but the second step is done using a fast multiplication technique for which only (12.3) holds? 11.5. (RESEARCH PROBLEM) Is one step of fixed precision iterative refinement sufficient to produce a componentwise relative backward error of order u for Cholesky factorization applied to a symmetric positive definite system Ax = b, assuming cond(A – 1 )σ(A , x) is not too large? Answer the same question for the diagonal pivoting method with partial pivoting applied to a symmetric system Ax = b.
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Chapter 12
Block LU Factorization
Block algorithms are advantageous for at least two important reasons. First, they work with blocks of data having b2 elements, performing O(b 3) operations. The O(b) ratio of work to storage means that processing elements with an O(b) ratio of computing speed to input/output bandwidth can be tolerated. Second, these algorithms are usually rich in matrix multiplication. This is an advantage because nearly every modern parallel machine is good at matrix multiplication. —ROBERT S. SCHREIBER, Block Algorithms for Parallel Machines (1988)
It should be realized that, with partial pivoting, any matrix has a triangular factorization. DECOMP actually works faster when zero pivots occur because they mean that the corresponding column is already in triangular form. — GEORGE E. FORSYTHE, MICHAEL A. MALCOLM, and CLEVE B. MOLER, Computer Methods for Mathematical Computations (1977)
It was quite usual when dealing with very large matrices to perform an iterative process as follows: the original matrix would be read from cards and the reduced matrix punched without more than a single row of the original matrix being kept in store at any one time; then the output hopper of the punch would be transferred to the card reader and the iteration repeated. — MARTIN CAMPBELL-KELLY, Programming the Pi/et ACE (1981)
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12.1. Block Versus Partitioned LU Factorization As we noted in Chapter 9 (Notes and References), Gaussian elimination (GE) comprises three nested loops that can be ordered in six ways, each yielding a different algorithmic variant of the method. These variants involve different computational kernels: inner product and saxpy operations (level-1 BLAS), or outer product and gaxpy operations (level-2 BLAS). To introduce matrix– matrix operations (level-3 BLAS), which are beneficial for high-performance computing, further manipulation beyond loop reordering is needed. We will use the following terminology, which emphasises an important distinction. A partitioned algorithm is a scalar (or point) algorithm in which the operations have been grouped and reordered into matrix operations. A block algorithm is a generalization of a scalar algorithm in which the basic scalar operations become matrix operations (a+ β, aβ, and a/β become A+B, AB, and AB-1), and a matrix property based on the nonzero structure becomes the corresponding property blockwise (in particular, the scalars 0 and 1 become the zero matrix and the identity matrix, respectively). A block factorization is defined in a similar way and is usually what a block algorithm computes. A partitioned version of the outer product form of LU factorization may be developed as follows. For and a given block size r, write (12.1) where A11 is r × r. One step of the algorithm consists of factoring A11 = L 11 U 11, solving the multiple right-hand side triangular systems L 11 U 12 = A1 2 and L2 1 U 11 = A21 for U 12 and L21, respectively, and then forming B = A22 – L 21 U 12; this procedure is repeated on B. The block operations defining U 12, L21, and B are level-3 BLAS operations. This partitioned algorithm does precisely the same arithmetic operations as any other variant of GE, but it does the operations in an order that permits them to be expressed as matrix operations. A genuine block algorithm computes a block LU factorization, which is a factorization A = LU where L and U are block triangular and L has identity matrices on the diagonal:
In general, the blocks can be of different dimensions. Note that this factorization is not the same as a standard LU factorization, because U is not
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triangular. However, the standard and block LU factorizations are related as follows: if A = LU is a block LU factorization and each Uii has LU factorization Uii = then A = Ldiag U is an LU factorization. Conditions for the existence of a block LU factorization are easy to state. Theorem 12.1. The matrix A = has a unique block LU factorization if and only if the first m – 1 leading principal block submatrices of A are nonsingular. Proof. The proof is entirely analogous to the proof of Theorem 9.1. This theorem makes clear that a block LU factorization may exist when an LU factorization does not. is nonsingular we can write If A 1 1 (12.2) which describes one block step of an outer-product-based algorithm for computing a block LU factorization. Here, S = A 22 – is the Schur complement of A11 in A. If the (1, 1) block of S of appropriate dimension is nonsingular then we can factorize S in a similar manner, and this process can be continued recursively to obtain the complete block LU factorization. The overall algorithm can be expressed as follows. Algorithm 12.2 (block LU factorization). This algorithm computes a block LU factorization A = LU using the notation of (12.2). 1. U 11 = A11, U 12 = A1 2 . 2. Solve L2 1 A 11 = A21 for L2 1 . 3. S = A22 – L2 1 A 12 (Schur complement). 4. Compute the block LU factorization of S, recursively. Given a block LU factorization of A, the solution to a system Ax = b can be obtained by solving Ly = b by forward substitution (since L is triangular) and solving Ux = y by block back substitution. There is freedom in how step 2 of Algorithm 12.2 is accomplished, and how the linear systems with coefficient matrices U ii that arise in the block back substitution are solved. The two main possibilities are as follows. Implementation 1: A11 is factorized by GE with partial pivoting. Step 2 and the solution of linear systems with Uii are accomplished by substitution with the LU factors of A1 1 . Implementation 2: is computed explicitly, so that step 2 becomes a matrix multiplication and Ux = y is solved entirely by matrix–vector multiplications. This approach is attractive for parallel machines.
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What can be said about the numerical stability of partitioned and block LU factorization? Because the partitioned algorithm is just a rearrangement of standard GE, the standard error analysis applies if the matrix operations are computed in the conventional way. However, if fast matrix multiplication techniques are used (for example, Strassen’s method), the standard results are not applicable. Standard results are, in any case, not applicable to block LU factorization; its stability can be very different from that of LU factorization. Therefore we need error analysis for both partitioned and block LU factorization based on general assumptions that permit the use of fast matrix multiplication. Unless otherwise stated, in this chapter an unsubscripted norm denotes ||A|| := maxi,j |aij|. We make two assumptions about the underlying level-3 BLAS (matrix-matrix operations). then the computed approximation to (1) If and C = AB satisfies (12.3) where c1 (m, n, p) denotes a constant depending on m, n and p. (2) The computed solution to the triangular systems TX = B, where and satisfies (12.4) For conventional multiplication and substitution, conditions (12.3) and (12.4) hold with c1 (m, n, p) = n 2 and c2 (m,p) = m2. For implementations based on Strassen’s method, (12.3) and (12.4) hold with c1 and c2 rather complicated functions of the dimensions m, n, p and the threshold no that determines the level of recursion (see Theorem 22.2 and [544, 1990]).
12.2. Error Analysis of Partitioned LU Factorization An error analysis for partitioned LU factorization must answer two questions. The first is whether partitioned LU factorization becomes unstable in some fundamental way when fast matrix multiplication is used. The second is whether the constants in (12.3) and (12.4) are propagated stably into the final error bound (exponential growth of the constants would be disastrous). We will assume that the block level LU factorization is done in such a way that the computed LU factors of A1 1 satisfy (12.5) Theorem 12.3 (Demmel and Higham). Under the assumptions (12.3), (12.4), and (12.5), the LU factors of computed using the partitioned
12.2 E RROR A NALYSIS
OF
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outer product form of LU factorization with block size r satisfiy where
249 = A+ ∆A, (12.6)
and where
Proof. The proof is essentially inductive. To save clutter we will omit “+O(u 2)” from each bound. For n = r, the result holds trivially. Consider the first block stage of the factorization, with the partitioning (12.1). The assumptions imply that (12.7) (12.8) To obtain B = A22 – L2 1 U12 we first compute C =
obtaining
and then subtract from A22, obtaining (12.9) It follows that (12.10a) (12.10b) The remainder of the algorithm consists of the computation of the LU factorization of B, and by our inductive assumption (12.6), the computed LU factors satisfy (12.11a) (12.11b) Combining (12.10) and (12.11), and bounding
using (12.9), we obtain
(12.12)
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Collecting (12.5), (12.7), (12.8), and (12.12) we have = A + ∆A, where bounds on ||∆Aij|| are given in the equations just mentioned. These bounds for the blocks of ∆A can be weakened slightly and expressed together in the more succinct form (12.6). These recurrences for δ(n,r) and θ(n,r) show that the basic error constants in assumptions (12.3), (12.4), and (12.5) combine additively at worst. Thus, the backward error analysis for the LU factorization is commensurate with the error analysis for the particular implementation of the BLAS3 employed in the partitioned factorization. In the case of the conventional BLAS3 we obtain a Wilkinson-style result for GE without pivoting, with θ(n,r) = O(n 3 ) (the growth factor is hidden in and Although the above analysis is phrased in terms of the partitioned outer product form of LU factorization, the same result holds for other “ijk” partitioned forms (with slightly different constants), for example, the gaxpy or sdot forms. There is no difficulty in extending the analysis to cover partial pivoting and solution of Ax = b using the computed LU factorization (see Problem 12.6).
12.3. Error Analysis of Block LU Factorization Now we turn to block LU factorization. We assume that the computed matrices L21 from step 2 of Algorithm 12.2 satisfy
We also assume that when a system U i i x i = d i of order r is solved, the computed solution satisfies (12.14) The assumptions (12.13) and (12.14) are satisfied for Implementation 1 of Algorithm 12.2 and are sufficient to prove the following result. Theorem 12.4 (Demmel, Higham, and Schreiber). Let and be the computed block LU factors of from Algorithm 12.2 (with Implementation 1), and let be the computed solution to Ax = b. Under the assumptions (12.3), (12.13), and (12.14),
(12.15) where the constant dn is commensurate with those in the assumptions.
12.3 E RROR A NALYSIS
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Proof. We omit the proof (see Demmel, Higham, and Schreiber [293, for details). It is similar to the proof of Theorem 12.3.
1995]
The bounds in Theorem 12.4 are valid also for other versions of block LU factorization obtained by “block loop reordering”, such as a block gaxpy based algorithm. Theorem 12.4 shows that the stability of block LU factorization is determined by the ratio ||L||||U||/||A|| (numerical experiments show that the bounds are, in fact, reasonably sharp). If this ratio is bounded by a modest function of n, then L and U are the true factors of a matrix close to A, and solves a slightly perturbed system. However, can exceed ||A|| by an arbitrary factor, even if A is symmetric positive definite or diusing the agonally dominant by rows. Indeed, ||L|| > ||L2 1 || = partitioning (12.2), and this lower bound for ||L|| can be arbitrarily large. In the following two subsections we investigate this instability more closely and show that ||L||||U|| can be bounded in a useful way for particular classes of A. Without further comment we make the reasonable assumption that ||L||U|| ||L||||U|| , so that these bounds maybe used in Theorem 12.4. What can be said for Implementation 2? Suppose, for simplicity, that the inverses (which are used in step 2 of Algorithm 12.2 and in the block back substitution) are computed exactly. Then the best bounds of the forms (12.13) and (12.14) are
Working from these results, we find that Theorem 12.4 still holds provided the first-order terms in the bounds in (12.15) are multiplied by max i κ(U i i ). This suggests that Implementation 2 of Algorithm 12.2 can be much less stable than Implementation 1 when the diagonal blocks of U are ill conditioned, and this is confirmed by numerical experiments.
12.3.1. Block Diagonal Dominance One class of matrices for which block LU factorization has long been known to be stable is block tridiagonal matrices that are diagonally dominant in an appropriate block sense. A general matrix is block diagonally dominant by columns with respect to a given partitioning A = ( A i j ) and a given norm if, for all j, (12.16) A is block diagonally dominant by rows if AT is block diagonally dominant by columns. For the block size 1, the usual property of point diagonal dominance
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is obtained. Note that for the 1- and co-norms diagonal dominance does not imply block diagonal dominance, nor does the reverse implication hold (see Problem 12.2). Throughout our analysis of block diagonal dominance we take the norm to be an arbitrary subordinate matrix norm. First, we show that for block diagonally dominant matrices a block LU factorization exists, using the key property that block diagonal dominance is inherited by the Schur complements obtained in the course of the factorization. In the analysis we assume that A has m block rows and columns. Theorem 12.5 (Demmel, Higham, and Schreiber). Suppose is nonsingular and block diagonally dominant by rows or columns with respect to a subordinate matrix norm in (12.16). Then A has a block LU factorization, and all the Schur complements arising in Algorithm 12.2 have the same kind of diagonal dominance as A. Proof. This proof is a generalization of Wilkinson’s proof of the corresponding result for point diagonally dominant matrices [1085, 1961, pp. 288– 289], [470, 1989, p. 120] (as is the proof of Theorem 12.6 below). We consider the case of block diagonal dominance by columns; the proof for row-wise diagonal dominance is analogous. The first step of Algorithm 12.2 succeeds, since A 11 is nonsingular, producing a matrix that we can write as
For j = 2:m we have
using (12.16)
using (12.16),
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OF
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(12.17) Now if is singular it follows that hence also A, is singular, which is a contradiction. Thus and (12.17) can be rewritten
therefore A(2), and is nonsingular,
showing that A(2) is block diagonally dominant by columns. The result follows by induction. The next result allows us to bound ||U|| for a block diagonally dominant matrix. Theorem 12.6 (Demmel, Higham, and Schreiber). Let A satisfy the conditions of Theorem 12.5. If A(k) denotes the matrix obtained after k – 1 steps of Algorithm 12.2, then
Proof. Let A be block diagonally dominant by columns (the proof for row diagonal dominance is similar). Then
using (12.16). By induction, using Theorem 12.5, it follows that This yields
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The implications of Theorems 12.5 and 12.6 for stability are as follows. Suppose A is block diagonally dominant by columns. Also, assume for the moment that the (subordinate) norm has the property that (12.18) which holds for any p-norm, for example. The subdiagonal blocks in the first block column of L are given by Li1 = and so < 1, by (12.16) and (12.18). From Theorem 12.5 it follows that < 1 for j = 2:m. Since Uij = for j > i, Theorem 12.6 shows that ||Uij|| < 2||A|| for each block of U (and ||Uij|| < ||A||). Therefore ||L|| < m and ||U|| < m2 ||A||, and so ||L||||U|| < m 3 ||A|| . For particular norms the bounds on the blocks of L and U yield a smaller bound for ||L|| and ||U||. For example, for the 1-norm we have ||L||1 ||U||1 < 2m||A||1 and for the -norm We conclude that block LU factorization is stable if A is block diagonally dominant by columns with respect to any subordinate matrix norm satisfying (12.18). Unfortunately, block LU factorization can be unstable when A is block diagonally dominant by rows, for although Theorem 12.6 guarantees that ||U i j || < 2||A||, ||L|| can be arbitrarily large. This can be seen from the example
where A is block diagonally dominant by rows in any subordinate norm for any nonsingular matrix A11. It is easy to confirm numerically that block LU factorization can be unstable on matrices of this form. Next, we bound ||L||||U|| for a general matrix and then specialize to point diagonal dominance. From this point on we use the norm ||A|| := max i,j |aij|. We partition A according to (12.19) and denote by pn the growth factor for GE without pivoting. We assume that GE applied to A succeeds. To bound ||L||, we note that, under the partitioning (12.19), for the first block stage of Algorithm 12.2 we have ||L2 1 || = < n pn κ(A) (see Problem 12.4). Since the algorithm works recursively with the Schur complement S, and since every Schur complement satisfies κ( S) < pn κ(A) (see Problem 12.4), each subsequently computed subdiagonal block of L has norm at most Since U is composed of elements of A together with elements of Schur complements of A, ||U|| < pn||A||.
(12.20)
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Overall, then, for a general matrix (12.21) Thus, block LU factorization is stable for a general matrix A as long as GE is stable for A (that is, pn is of order 1) and A is well conditioned. If A is point diagonally dominant by columns then, since every Schur complement enjoys the same property, we have ||Lij || < 1 for i > j, by Problem 12.5. Hence ||L|| = 1. Furthermore, pn < 2 (Theorem 9.8 or Theorem 12.6), giving ||U|| < 2||A|| by (12.20), and so ||L||||U|| < 2||A||. Thus block LU factorization is perfectly stable for a matrix point diagonally dominant by columns. If A is point diagonally dominant by rows then the best we can do is to take pn < 2 in (12.21), obtaining ||L||||U|| < 8nk(A)||A||.
(12.22)
Hence for point row diagonally dominant matrices, stability is guaranteed if A is well conditioned. This in turn is guaranteed if the row diagonal dominance amounts γj in the analogue of (12.16) for point row diagonal dominance are sufficiently large relative to ||A||, because < (minj γj )–1 (see problem 8.7(a)). 12.3.2. Symmetric Positive Definite Matrices Further useful results about the stability of block LU factorization can be derived for symmetric positive definite matrices. First, note that the existence of a block LU factorization is immediate for such matrices, since all their leading principal submatrices are nonsingular. Let A be a symmetric positive definite matrix, partitioned as
The definiteness implies certain relations among the submatrices Aij that can be used to obtain a stronger bound for ||L|| 2 than can be deduced for a general matrix (cf. Problem 12.4). Lemma 12.7. If A is symmetric positive definite then Proof. This lemma is a corollary of Lemma 10.12, but we give a separate proof. Let A have the Cholesky factorization
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Table 12.1. Stability of block and point L U factorization. p n is the growth factor for GE without pivoting. Matrix property Symmetric positive definite Block column diagonally dominant Point column diagonally dominant Block row diagonally dominant Point row diagonally dominant Arbitrary
Block LU Point LU κ(A) 1 / 2 1 1 pn 1 1 pn κ(A) 1 pn
The following lemma is proved in a way similar to the second inequality in Problem 12.4. Lemma 12.8. If A is symmetric positive definite then the Schur complement S = A 22 – satisfies κ2 (S) < κ 2 (A). Using the same reasoning as in the last subsection, we deduce from these two lemmas that each subdiagonal block of L is bounded in 2-norm by κ 2 (A) 1 / 2 . Therefore ||L|| 2 < 1 + mκ 2 (A)1/2, where there are m block stages in the algorithm. Also, it can be shown that ||U|| 2 < Hence (12.23) It follows from Theorem 12.4 that when Algorithm 12.2 is applied to a symmetric positive definite matrix A, the backward errors for the LU factorization and the subsequent solution of a linear system are both bounded by (12.24) Any resulting bound for will be proportional to κ 2 (A)3/2, rather than κ2 (A) as for a stable method. This suggests that block LU factorization can lose up to 50% more digits of accuracy in x than a stable method for solving symmetric positive definite linear systems. The positive conclusion to be drawn, however, is that block LU factorization is guaranteed to be stable for a symmetric positive definite matrix that is well conditioned. The stability results for block LU factorization are summarized in Table 12.1, which tabulates a bound for ||A – LU||/(cnu||A||) for block and point
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LU factorization for the matrix properties considered in this chapter. The constant cn incorporates any constants in the bound that depend polynomially on the dimension, so a value of 1 in the table indicates unconditional stability.
12.4. Notes and References The distinction between a partitioned algorithm and a block algorithm is rarely made in the literature (exceptions include the papers by Schreiber [902, 1988] and Demmel, Higham, and Schreiber [293, 1995]); the term “block algorithm” is frequently used to describe both types of algorithm. A partitioned algorithm might also be called a “blocked algorithm” (as is done by Dongarra, Duff, Sorensen, and van der Vorst [315, 1991]), but the similarity of this term to “block algorithm” can cause confusion and so we do not recommend this terminology. Note that in the particular case of matrix multiplication, partitioned and block algorithms are equivalent. Our treatment of partitioned LU factorization has focused on the stability aspects; for further details, particularly concerning implementation on high-performance computers, see Dongarra, Duff, Sorensen, and van der Vorst [315, 1991] and Golub and Van Loan [470, 1989]. Block LU factorization appears to have first been proposed for block tridiagonal matrices, which frequently arise in the discretization of partial differential equations. References relevant to this application include Isaacson and Keller [607, 1966, p. 59], Varah [1048, 1972], Bank and Rose [53, 1977], Mattheij [737, 1984], [738, 1984], and Concus, Golub, and Meurant [235, 1985]. For an application of block LU factorization to linear programming, see Eldersveld and Saunders [351, 1992]. Theorem 12.3 is from Demmel and Higham [291, 1992]. The results in §12.3 are from Demmel, Higham, and Schreiber [293, 1995], which extends earlier analysis of block LU factorization by Demmel and Higham [291, 1992]. Block diagonal dominance was introduced by Feingold and Varga [366, 19 6 2 ], and has been used mainly in generalizations of the Gershgorin circle theorem. Varah [1048, 1972] obtained bounds on ||L|| and ||U|| for block diagonally dominant block tridiagonal matrices; see Problem 12.1. Theorem 12.5 is obtained in the case of block diagonal dominance by rows with minj γ j > 0 by Polman [837, 1987]; the proof in [837, 1987] makes use of the corresponding result for point diagonal dominance and thus differs from the proof we have given. At the cost of a much more difficult proof, Lemma 12.7 can be strengthened to the attainable bound < (κ 2 ( A)1/2 – κ 2 (A)-1/2)/2, as shown by Demmel [279, 1983, Thin. 4], but the weaker bound is sufficient for our purposes.
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258 12.4.1. LAPACK
LAPACK does not implement block LU factorization, but its LU factorization (and related) routines for full matrices employ partitioned LU factorization in order to exploit the level-3 BLAS and thereby to be efficient on highperformance machines.
Problems 12.1. (Varah [1048, 1972]) Suppose A is block tridiagonal and has the block LU factorization A = LU (so that L and U are block bidiagonal and Ui,i + 1 = A i , i +1). Show that if A is block diagonally dominant by columns then
while if A is block diagonally dominant by rows then
What can be deduced about the stability of the factorization and has the block classes of matrices? 12.2. Show that for the 1- and co-norms diagonal dominance does not imply block diagonal dominance, and vice versa. is symmetric, has positive diagonal elements, and is block 12.3. If diagonally dominant by rows, must it be positive definite? 12.4. Let
be partitioned (12.25)
with A11 nonsingular. Let ||A|| := max ij |aij|. Show that n p n κ(A). Show that the Schur complement S = A 22 – κ (S) < pn κ(A).
satisfies
12.5. Let be partitioned as in (12.25), with A11 nonsingular, and suppose that A is point diagonally dominant by columns. Show that < 1. 12.6. Show that under the conditions of Theorem 12.3 the computed solution to Ax = b satisfies
PROBLEMS
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and the computed solution to the multiple right-hand side system AX = B (where (12.4) is assumed to hold for the multiple right-hand side triangular solves) satisfies
In both cases, cn is a constant depending on n and the block size. 12.7. Let X = that
where A is square and nonsingular. Show det(X) = det(A) det(D – CA- 1 B ) .
Assuming A, B, C, D are all m × m, give a condition under which det(X ) = det( AD – CB).
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Chapter 13
Matrix Inversion
It is amusing to remark that we were so involved with matrix inversion that we probably talked of nothing else for months. Just in this period Mrs. von Neumann acquired a big, rather wild but gentle Irish Setter puppy, which she called inverse in honor of our work! — HERMAN H. GOLDSTINE, The Computer.’ From Pascal to von Neumann (1972)
The most computationally intensive portion of the tasks assigned to the processors is integrating the KKR matrix inverse over the first Brillouin zone. To evaluate the integral, hundreds or possibly thousands of complex double precision matrices of order between 80 and 300 must be formed and inverted. Each matrix corresponds to a different vertex of the tetrahedrons into which the Brillouin zone has been subdivided. — M. T. HEATH, G. A. GEIST, and J. B. DRAKE, Superconductivity in Early Experience with the Intel iPSC/860 at Oak Ridge National Laboratory (1990) Press to invert the matrix. Note that matrix inversion can produce erroneous results if you are using iii-conditioned matrices. — HEWLETT-PACKARD, HP 48G Series User’s Guide (1993)
Almost anything you can do with A–1 can be done without it. — GEORGE E. FORSYTHE and CLEVE B. MOLER, Computer Solution of Linear Algebraic Systems (1967)
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13.1. Use and Abuse of the Matrix Inverse To most numerical analysts, matrix inversion is a sin. Forsythe, Malcolm, and Moler put it well when they say [395, 1977, p. 31] “In the vast majorit y of practical computational problems, it is unnecessary and inadvisable to actually compute A– 1 .” The best example of a problem in which the matrix inverse should not be computed is the linear equations problem Ax = b. Computing the solution as x = A-1 × b requires 2n 3 flops, assuming A - 1 is computed by Gaussian elimination with partial pivoting (GEPP), whereas GEPP applied directly to the system costs only 2n3/3 flops. Not only is the inversion approach three times more expensive, but it is much less stable. Suppose X = A–1 is formed exactly, and that the only rounding errors are in forming x = fl(Xb). Then = (X + ∆X) b , where |∆X| < γn|X|, by (3.10). So = A(X + ∆X)b = (I + A∆X)b, and the best possible residual bound is
For GEPP, Theorem 9.4 yields
Since it is usually true that for GEPP, we see that the matrix inversion approach is likely to give a much larger residual than GEPP if A is ill conditioned and if For example, we solved 50 25×25 systems Ax = b in MATLAB, where the elements of x are taken from the normal N(0, 1) distribution and A is random with κ 2 (A) = u - 1 / 2 9 × 107. As Table 13.1 shows, the inversion approach provided much larger backward errors than GEPP in this experiment. Given the inexpedience of matrix inversion, why devote a chapter to it? The answer is twofold. First, there are situations in which a matrix inverse must be computed. Examples are in statistics [54, 1974, §7.5], [721, 198 4, §2.3], [744, 1989, p. 342 ff], where the inverse can convey important statistical information, in certain matrix iterations arising in eigenvalue-related problems [37, 1993], [174, 198 7], [566, 1994], and in numerical integrations arising in
Table 13.1. Backward errors η A , b min x=A
-1
× b GEPP
for the max
6.66e-12 1.69e-10 3.44e-18 7.56e-17
-norm.
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AND
ABUSE
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superconductivity computations [509, 1990] (see the quotation at the start of the chapter). Second, methods for matrix inversion display a wide variety of stability properties, making for instructive and challenging error analysis. (Indeed, the first major rounding error analysis to be published, that of von Neumann and Goldstine, was for matrix inversion-see §9.6). Matrix inversion can be done in many different ways—in fact, there are more computationally distinct possibilities than for any other basic matrix computation. For example, in triangular matrix inversion different loop orderings are possible and either triangular matrix–vector multiplication, solution of a triangular system, or a rank-1 update of a rectangular matrix can be employed inside the outer loop. More generally, given a factorization PA = LU, two ways to evaluate A–1 are. as A–1 = U –1 × L –1 × P, and as the solution to UA–1 = L –1 × P. These methods generally achieve different levels of efficiency on high-performance computers, and they propagate rounding errors in different ways. We concentrate in this chapter on the numerical stability, but comment briefly on performance issues. The quality of an approximation Y A–1 can be assessed by looking at the right and left residuals, AY — I and YA — I , and the forward error, Y – A-1. Suppose we perturb A A + ∆A with |∆ A| < thus, we are making relative perturbations of size at most to the elements of A. If Y = (A+ ∆A)-1 then (A + ∆A)Y = Y(A + ∆A) = I, so that (13.1) (13.2) and, since (A + ∆A)–1 = A–1 – A– 1 ∆ AA–1 + (13.3) (Note that (13.3) can also be derived from (13.1) or (13.2 ).) The bounds (13.1)-(13.3) represent “ideal” bounds for a computed approximation Y to A – 1, if we regard as a small multiple of the unit roundoff u. We will show that, for triangular matrix inversion, appropriate methods do indeed achieve (13.1) or (13.2) (but not both) and (13.3). It is important to note that neither (13.1), (13.2), nor (13.3) implies that Y + ∆Y = (A + ∆A)-1 with and that is, Y need not be close to the inverse of a matrix near to A, even in the norm sense. Indeed, such a result would imply that both the left and right residuals are bounded in norm by and this is not the case for any of the methods we will consider. To illustrate the latter point we give a numerical example. Define the matrix An as triu(qr(vand(n))), in M ATLAB notation (vand is a routine from the Test Matrix Toolbox—see Appendix E); in other words, An is the upper triangular QR factor of the n × n Vandermonde matrix based on
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Figure 13.1. Residuals for inverses computed by MATLAB’S INV function. equispaced points on [0, 1]. We inverted An , for n = 1:80, using MATLAB’S INV function, which uses GEPP. The left and right normwise relative residuals
are plotted in Figure 13.1. We see that while the left residual is always less than the unit roundoff, the right residual becomes large as n increases. These matrices are very ill conditioned (singular to working precision for n > 20), yet it is still reasonable to expect a small residual, and we will prove in §13.3.2 that the left residual must be small, independent of the condition number. In most of this chapter we are not concerned with the precise values of constants (§13.4 is the exception); thus cn denotes a constant of order n. To simplify the presentation we introduce a special notation. Let Ai i = 1:k, be matrices such that the product A 1 A 2 . . . Ak is defined and let
Then ∆(A1, A2, . . . , Ak)
denotes a matrix bounded according to
13.2 I NVERTING
A
T RIANGULAR M ATRIX
This notation is chosen so that if evaluated in any order, then
265
= fl(A 1 A 2 . . .A k), with the product
13.2. Inverting a Triangular Matrix treating We consider the inversion of a lower triangular matrix unblocked and blocked methods separately. We do not make a distinction between partitioned and block methods in this section. All the results in this and the next section are from Du Croz and Higham [322, 1992]. 13.2.1. Unblocked Methods We focus our attention on two “j” methods that compute L–1 a column at a time. Analogous “i” and “k” methods exist, which compute L–1 row-wise or use outer products, respectively, and we comment on them at the end of the section. The first method computes each column of X = L-1 independently, using forward substitution. We write it as follows, to facilitate comparison with the second method. Method 1. for j = 1:n X(j + 1:n,j) = -x jj L(j + 1:n,j) Solve L(j + 1:n, j + l:n)X(j + 1:n, j) = X(j + 1:n,j), by forward substitution. end In BLAS terminology, this method is dominated by n calls to the level-2 BLAS routine xTRSV (Triangular SolVe). The second method computes the columns in the reverse order. On the jth step it multiplies by the previously computed inverse L(j + 1:n, j + 1:n) - l instead of solving a system with coefficient matrix L(j + 1:n, j + 1:n). Method 2. for j = n:–1:1 X (j + 1:n, j) = X(j + 1:n, j + 1:n)L(j + 1:n, j) X (j + 1:n, j) = -xjjX(j + 1:n, j) end
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Method 2 uses n calls to the level-2 BLAS routine xTRMV (Triangular Matrix times Vector). On most high-performance machines xTRMV can be implemented to run faster than xTRSV , so Method 2 is generally preferable to Method 1 from the point of view of efficiency (see the performance figures at the end of §13.2.2). We now compare the stability of the two methods. Theorem 8.5 shows that the jth column of the computed from Method 1 satisfies It follows that we have the componentwise residual bound (13.4) and the componentwise forward error bound (13.5) Since
= L-1 + O(u), (13.5) can be written as (13.6)
which is invariant under row and column scaling of L. If we take norms we obtain normwise relative error bounds that are either row or column scaling independent: from (13.6) we have (13.7) and the same bound holds with cond(L –1) replaced by cond(L). Notice that (13.4) is a bound for the right residual, LX – I. This is because Method 1 is derived by solving LX = I. Conversely, Method 2 can be derived by solving XL = I, which suggests that we should look for a bound on the left residual for this method. Lemma 13.1. The computed inverse
from Method 2 satisfies (13.8)
Proof. The proof is by induction on n, the case n = 1 being trivial. Assume the result is true for n – 1 and write
where and the first column of X by solving XL = I according to β = a- 1 ,
z = −βN y.
Method 2 computes
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In floating point arithmetic we obtain
Thus
This may be written as
By assumption, the corresponding inequality holds for the (2:n , 2:n ) submatrices and so the result is proved. Lemma 13.1 shows that Method 2 has a left residual analogue of the right residual bound (13.4) for Method 1. Since there is, in general, no reason to choose between a small right residual and a small left residual, our conclusion is that Methods 1 and 2 have equally good numerical stability properties. More generally, it can be shown that all three i, j, and k inversion variants that can be derived from the equations LX = I produce identical rounding errors under suitable implementations, and all satisfy the same right residual bound; likewise, the three variants corresponding to the equation XL = I all satisfy the same left residual bound. The LINPACK routine xTRDI uses a k variant derived from XL = I; the LINPACK routines xGEDI and xPODI contain analogous code for inverting an upper triangular matrix (but the LINPACK Users’ Guide [307, 1979, Chaps. 1 and 3] describes a different variant from the one used in the code). 13.2.2. Block Methods Let the lower triangular matrix
be partitioned in block form as
(13.9)
where we place no restrictions on the block sizes, other than to require the diagonal blocks to be square. The most natural block generalizations of Methods 1 and 2 are as follows. Here, we use the notation Lp:q,r:s to denote the
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submatrix comprising the intersection of block rows p to q and block columns r to s of L. Method 1B. for j = 1:N (by Method 1) X j+ 1 :N,j = -Lj+ 1 :N , j X j j Solve L j+ 1 :N , j+ 1 :N X j+ 1 :N,j = by forward substitution
Xj+ 1 :N , j,
end Method 2B. f o r j = N:–1:1 (by Method 2) X j+ 1 :N,j = Xj+ 1 :N , j+ 1 :N L j+ 1 :N , j X j+ 1 :N,j = -Xj+ 1 :N , j X jj end One can argue that Method 1B carries out the same arithmetic operations as Method 1, although possibly in a different order, and that it therefore satisfies the same error bound (13.4). For completeness, we give a direct proof. Lemma 13.2. The computed inverse
from Method 1B satisfies (13.10)
Proof. Equating block columns in (13.10), we obtain the N independent inequalities (13.11) It suffices to verify the inequality with j = 1. Write
and L11 is the (1, 1) block in the partitioning of (13.9). where L 11, X1 1 X 11 is computed by Method 1 and so, from (13.4), (13.12) X 21 is computed by forming T = –L2 1 X 11 and solving L2 2 X 21 = T. The computed X 21 satisfies
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Hence
(13.13) Together, inequalities (13.12) and (13.13) are equivalent to (13.11) with j = 1, as required. We can attempt a similar analysis for Method 2B. With the same notation as above, X21 is computed as X21 = –X 2 2L 2 1X 11. Thus (13.14) To bound the left residual we have to postmultiply by L 11 and use the fact that X11 is computed by Method 2:
This leads to a bound of the form
which would be of the desired form in (13.8) were it not for the factor This analysis suggests that the left residual is not guaranteed to be small. Numerical experiments confirm that the left and right residuals can be large simultaneously for Method 2B, although examples are quite hard to find [322, 1992]; therefore the method must be regarded as unstable when the block size exceeds 1. The reason for the instability is that there are two plausible block generalizations of Method 2 and we have chosen an unstable one that does not carry out the same arithmetic operations as Method 2. If we perform a solve with Ljj instead of multiplying by Xjj we obtain the second variation, which is used by LAPACK’s xTRTRI : Method 2C. for j = N:-1:1 (by Method 2) X j+ 1 :N,j = Xj+ 1 :N , j+ 1 :N L j+ 1 :N , j Solve Xj+ 1 :N,jLjj = –Xj+ 1 :N,j by back substitution. end For this method, the analogue of (13.14) is
which yields
Hence Method 2C enjoys a very satisfactory residual bound.
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Table 13.2. Mflop rates for inverting a triangular matrix on a Cray 2.
Method 1 Method 2 k variant Blocked: Method 1B (block size 64) Method 2C k variant Unblocked:
n = 128 n = 256 n = 512 n = 1024 162 231 95 283 114 211 330 289 157 178 114 191 125 246 348 405 129 269 428 378 148 344 263 383
Lemma 13.3. The computed inverse
from Method 2C satisfies
In summary, block versions of Methods 1 and 2 are available that have the same residual bounds as the point methods. However, in general, there is no guarantee that stability properties remain unchanged when we convert a point method to block form, as shown by Method 2B. In Table 13.2 we present some performance figures for inversion of a lower triangular matrix on a Cray 2. These clearly illustrate the possible gains in efficiency from using block methods, and also the advantage of Method 2 over Method 1. For comparison, the performance of a k variant is also shown (both k variants run at the same rate). The performance characteristics of the i variants are similar to those of the j variants, except that since they are row oriented rather than column oriented, they are liable to be slowed down by memory-bank conflicts, page thrashing, or cache missing.
13.3. Inverting a Full Matrix by LU Factorization Next, we consider four methods for inverting a full matrix given an LU factorization computed by GEPP. We assume, without loss of generality, that there are no row interchanges. We write the computed LU factors as L and U. Recall that A + ∆A = LU, with |∆ A| < cnu|L||U| (Theorem 9.3). 13.3.1. Method A Perhaps the most frequently described method for computing X = A-1 is the following one. Method A. for j = 1:n
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Solve AxJ = ej end Compared with the methods to be described below, Method A has the disadvantages of requiring more temporary storage and of not having a convenient partitioned version. However, it is simple to analyse. From Theorem 9.4 we have (13.15) and so (13.16) This bound departs from the form (13. 1) only in that |A| is replaced by its upper bound |L||U| + O(u). The forward error bound corresponding to (13.16) is (13.17) Note that (13.15) says that is the jth column of the inverse of a matrix close to A, but it is a different perturbation ∆Aj for each column. It is not true that itself is the inverse of a matrix close to A, unless A is well conditioned. 13.3.2. Method B Next, we consider the method used by LINPACK’S xGEDI , LAPACK’S xGETRI, and MATLAB’S INV function. Method B. Compute U -1 and then solve for X the equation XL = U - 1 . To analyse this method we will assume that U -1 is computed by an analogue of Method 2 or 2C for upper triangular matrices that obtains the columns of U –1 in the order 1 to n. Then the computed inverse Xu U– 1 will satisfy the residual bound
We also assume that the triangular solve from the right with L is done by back substitution. The computed X therefore satisfies XL = Xu + ∆(X, L) and so This leads to the residual bound
(13.18)
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which is the left residual analogue of (13.16). From (13.18) we obtain the forward error bound
Note that Methods A and B are equivalent, in the sense that Method A solves for X the equation LUX = I while Method B solves XLU = I. Thus the two methods carry out analogous operations but in different orders. It follows that the methods must satisfy analogous residual bounds, and so (13.18) can be deduced from (13.16). We mention in passing that the LINPACK manual states that for Method B a bound holds of the form ||AX – I|| < dnu||A|| ||X|| [307, 1979, p. 1.20]. This is incorrect, although counterexamples are rare; it is the left residual that is bounded this way, as follows from (13.18). 13.3.3. Method C The next method that we consider is from Du Croz and Higham [322, 1992]. It solves the equation UXL = I, computing X a partial row and column at a time. To derive the method partition
where the (1, 1) blocks are scalars, and assume that the trailing submatrix X 22 is already known. Then the rest of X is computed according to
The method can also be derived by forming the product X = U -1 × L - 1 using the representation of L and U as a product of elementary matrices (and diagonal matrices in the case of U). In detail the method is as follows. Method C. for k = n:–1:1
end The method can be implemented so that X overwrites L and U, with the aid of a work vector of length n (or a work array to hold a block row or column
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in the partitioned case). Because most of the work is performed by matrix– vector (or matrix-matrix) multiplication, Method C is likely to be the fastest of those considered in this section on many machines. (Some performance figures are given at the end of the section.) A straightforward error analysis of Method C shows that the computed satisfies (13.19) We will refer to as a “mixed residual”. From (13.19) we can obtain bounds on the left and right residual that are weaker than those in (13.18) and (13.16) by a factor |U- 1 ||U| on the left or |L||L - 1| on the right, respectively. We also obtain from (13.19) the forward error bound
which is (13.17) with |A- 1 | replaced by its upper bound |U – 1 ||L - 1 | + O(u) and the factors reordered. The LINPACK routine xSIDI uses a special case of Method C in conjunction with the diagonal pivoting method to invert a symmetric indefinite matrix; see Du Croz and Higham [322, 1992] for details. 13.3.4. Method D The next method is based on another natural way to form A -1 and is used by LAPACK’S xPOTRI , which inverts a symmetric positive definite matrix. Method D. Compute L-1 and U -1 and then form A-1 = U -1 × L- 1 . The advantage of this method is that no extra workspace is needed; U - 1 and L–1 can overwrite U and L, and can then be overwritten by their product. However, Method D is significantly slower on some machines than Methods B or C, because it uses a smaller average vector length for vector operations. To analyse Method D we will assume initially that L -1 is computed by Method 2 (or Method 2C) and, as for Method B above, that U -1 is computed by an analogue of Method 2 or 2C for upper triangular matrices. We have (13.20) Since A = LU – ∆A, (13.21) Rewriting the first term of the right-hand side using XLL = I + ∆ ( X L , L ) , and similarly for U, we obtain (13.22)
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and so
(13.23) This bound is weaker than (13.18), since + O(u). Note, however, that the term ∆(XU,XL)A in the residual (13.22) is an unavoidable consequence of forming XU XL, and so the bound (13.23) is essentially the best possible. The analysis above assumes that XL and XU both have small left residuals. If they both have small right residuals, as when they are computed using Method 1, then it is easy to see that a bound analogous to (13.23) holds for the right residual – I. If XL has a small left residual and XU has a small right residual (or vice versa) then it does not seem possible to derive a bound of the form (13.23). However, we have |X L L - I| = |L- 1 (L X L - I)L| < |L- 1||LXL - I||L|,
(13.24)
and since L is unit lower triangular with |lij| < 1, we have |( L - 1 ) ij| < 2n - 1 , which places a bound on how much the left and right residuals of XL can differ. Furthermore, since the matrices L from GEPP tend to be well conditioned and since our numerical experience is that large residuals tend to occur only for ill-conditioned matrices, we would expect the left and right residuals of XL almost always to be of similar size. We conclude that even in the “conflicting residuals” case, Method D will, in practice, usually satisfy (13.23) or its right residual counterpart, according to whether XU has a small left or right residual respectively. Similar comments apply to Method B when U –1 is computed by a method yielding a small right residual. These considerations are particularly pertinent when we consider Method D specialized to symmetric positive definite matrices and the Cholesky factorization A = R T R. Now A-1 is obtained by computing XR = R-1 and then forming A–1 = this is the method used in the LINPACK routine xPODI [307, 1979, Chap. 3]. If XR has a small right residual then has a small left residual, so in this application we naturally encounter conflicting residuals. Fortunately, the symmetry and definiteness of the problem help us to obtain a satisfactory residual bound. The analysis parallels the derivation of (13.23), so it suffices to show how to treat the term (cf. (13.21)), where R now denotes the computed Cholesky factor. Assuming RX R = I + ∆(R, XR), and using (13.24) with L replaced by R, we have
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Table 13.3. Mflop rates for inverting a full matrix on a Cray 2.
Method B Method C Method D Blocked: Method B (block size 64) Method C Method D Unblocked:
n = 64 118 125 70 142 144 70
n = 128 n = 256 229 310 314 235 267 166 259 353 264 363 178 306
n = 512 347 351 329 406 415 390
and From the inequality + O(u), it follows that
together with ||A||2 =
and thus overall we have a bound of the form
Since
and A are symmetric the same bound holds for the right residual.
13.3.5. Summary In terms of the error bounds, there is little to choose between Methods A, B, C, and D. Numerical results reported in [322, 1992] show good agreement with the bounds. Therefore the choice of method can be based on other criteria, such as performance and the use of working storage. Table 13.3 gives some performance figures for a Cray 2, covering both blocked (partitioned) and unblocked forms of Methods B, C, and D. On a historical note, Tables 13.4 and 13.5 give timings for matrix inversion on some early computing devices; times for two modern machines are given for comparison. The inversion methods used for the timings on the early computers in Table 13.4 are not known, but are probably methods from this section or the next.
13.4. Gauss–Jordan Elimination Whereas Gaussian elimination (GE) reduces a matrix to triangular form by elementary operations, Gauss–Jordan elimination (GJE) reduces it all the way
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Table 13.4. Times (minutes and seconds) for inverting an n × n matrix. Source for DEUCE, Pegasus, and Mark 1 timings: [181, 1981].
Pegasus DEUCE (English Electric) (Ferranti) 1956 1955 26s 20s 2m 37s 58s 7m 57s 3m 36s 17m 52s 7m 44s
n 8 16 24 32
Manchester Mark 1 1951 — — 8m 16m
HP 48G Calculator 1993 4s 18s 48s —
Sun SPARCstation ELC 1991 .004s .01s .02s .04s
Table 13.5. Additional timings for inverting an n × n matrix. Machine Aiken Relay Calculator IBM 602 Calculating Punch SEAC (National Bureau of Standards) Datatron IBM 704
Year 1948 1949 1954 1957 1957
n Time 38 59½ hours 10 8 hours 49 3 hours 8 0 ’ 2½ hours 115* 19m 30s
Reference [764, 1948] [1053, 1949] [1004, 1954] [753, 1957] [320, 1957]
a
Block tridiagonal matrix, using an inversion method designed for such matrices. asymmetric positive definite matrix.
to diagonal form. GJE is usually presented as a method for matrix inversion, but it can also be regarded as a method for solving linear equations. We will take the method in its latter form, since it simplifies the error analysis. Error bounds for matrix inversion are obtained by taking unit vectors for the right-hand sides. At the kth stage of GJE, all off-diagonal elements in the kth column are eliminated, instead of just those below the diagonal, as in GE. Since the elements in the lower triangle (including the pivots on the diagonal) are identical to those that occur in GE, Theorem 9.1 tells us that GJE succeeds if all the leading principal submatrices of A are nonsingular. With no form of pivoting GJE is unstable in general, for the same reasons that GE is unstable. Partial and complete pivoting are easily incorporated. Algorithm 13.4 (Gauss–Jordan elimination). This algorithm solves the linis nonsingular, by GJE with partial ear system Ax = b, where pivoting.
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for k = 1:n Find r such that |ark| = maxi>k |aik|. A(k, k:n) A(r, k:n), b(k) b(r) % Swap rows k and r. row_ind = [1:k – 1, k + 1:n] % Row indices of elements to eliminate. m = A(row_ind,k)/A(k,k) % Multipliers. A(row_ind,k:n) = A(row_ind,k:n) – m*A(k,k:n) b(row_ind) = b(row_ind) – m*b(k) end xi = bi /aii , i = 1:n Cost: n3 flops. The numerical stability properties of GJE are rather subtle and error analysis is trickier than for GE. An observation that simplifies the analysis is that we can consider the algorithm as comprising two stages. The first stage is identical to GE and forms Mn- 1 M n-2 . . . M1 A = U, where U is upper triangular. The second stage reduces U to diagonal form by elementary operations:
The solution x is formed as x = D - 1 Nn . . . N2 y, where y = Mn-1 . . . M2 b. The rounding errors in the first stage are precisely the same as those in GE, so it suffices to consider the second stage. We will assume, for simplicity, that there are no row interchanges. As with GE, this is equivalent to assuming that all row interchanges are done at the start of the algorithm. Define Uk+l = Nk. . .N2 U (so U2 = U) and note that Nk and Uk have the forms
where nk = [-u 1 k /u k k , . . . . -uk- 1 ,k / u k k ] T. The computed matrices obviously satisfy (13.25a) (13.25b) (to be precise, is defined as Nk but with with xk+1 = Nk . . . N2 y, we have
Similarly,
(13.26)
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278 Because of the structure of
∆ k , and fk , we have the useful property that
Without loss of generality, we now assume that the final diagonal matrix D is the identity (i.e., the pivots are all 1); this simplifies the analysis a little and has a negligible effect on the final bounds. Thus (13.25) and (13.26) yield (13.27) (13.28) Now
and, similarly, But defining ñk
we have
where X = U-1 + O(u) (by (13.27)). Hence
Combining (13.27) and (13.28) we have, for the solution of Ux = y ,
which gives the componentwise forward error bound (13.29)
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Table 13.6. Gauss-Jordan elimination for Ux = b. n ηU , b 16 2.0e-14 32 6.4e-10 64 1.7e-2
5.8e-11 7.6e-6 6.6e4
This is an excellent forward error bound: it says that the error in is no larger than the error we would expect for an approximate solution with a tiny componentwise relative backward error. In other words, the forward error is bounded in the sane way as for substitution. However, we have not, and indeed cannot, show that the method is backward stable. The best we can do is to derive from (13.27) and (13.28) the result (13.30a) (13.30b) (13.30C) using = U + O(u). These bounds show that has a normwise backward error bounded approximately by Hence the backward error can be guaranteed to be small only if U is well conditioned. This agrees with the comments of Peters and Wilkinson [828, 1975] that “the residuals corresponding to the Gauss-Jordan solution are often larger than those corresponding to back-substitution by a factor of order κ.” A numerical example helps to illustrate the results. We take U to be the upper triangular matrix with 1s on the diagonal and – 1s everywhere above the diagonal (U = U(1) from (8.2)). This matrix has condition number Table 13.6 reports the normwise relative backward error (see (7.2)) for b = U x, where x = e/3. Clearly, GJE is backward unstable for these matrices—the backward errors show a dependence on However, the relative distance between and the computed solution from substitution (not shown in the table) is less than which shows that the forward error is bounded by confirming (13.29). By bringing in the error analysis for the reduction to upper triangular form, we obtain an overall result for GJE. Theorem 13.5. Suppose GJE successfully computes an approximate solution to Ax = b, where A is nonsingular. Then (13.31) (13.32)
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is the factorization computed by GE.
Proof. For the first stage (which is just GE), we have A + ∆A1 = by Theorems 9.3 and 8.5. Using (13.30), we obtain
or A = b – r, where (13.33) The bounds (13.31) and (13.32) follow easily on using (13.30). Theorem 13.5 shows that the stability of GJE depends not only on the size of |L||U| (as in GE), but also on the condition of U. The term is an upper bound for and if this bound is sharp then the residual bound is very similar to that for LU factorization. Note that for partial pivoting we have The bounds in Theorem 13.5 have the pleasing property that they are invariant under row or column scaling of A, though of course if we are using partial pivoting then row scaling can change the pivots and alter the bound. As mentioned earlier, to obtain bounds for matrix inversion we simply take b to be each of the unit vectors in turn. For example, the residual bound becomes
For the special case of symmetric positive definite matrices, an informative normwise result follows from Theorem 13.5. We make the natural assumption that symmetry is exploited in the elimination. Corollary 13.6. Suppose GJE successfully computes an approximate solution to Ax = b, where is symmetric positive definite. Then
where
is the factorization computed by symmetric GE.
Proof. By Theorem 9.3 we gave A + ∆A = ic and satisfies |∆ A| < Defining D =
where ∆A is symmetrwe have, by
13.5 T HE D ETERMINANT symmetry, A + ∆A =
281 RTR. Hence
Furthermore, it is staightforward to show that n (1- γ n ) - 1 ||A|| 2 . The required bounds follow. Corollary 13.6 shows that GJE is forward stable for symmetric positive definite matrices, but it bounds the backward error only by a multiple of κ 2 (A)1/2. Numerical experiments show that the backward error is usually much less than κ 2 (A) 1 / 2 u, but (very ill-conditioned) matrices can certainly be found for which the backward error is many order of magnitude larger than u. Hence GJE is not backward stable even for symmetric positive definite matrices.
13.5. The Determinant It may be too optimistic to hope that determinants will fade out of the mathematical picture in a generation; their notation alone is a thing of beauty to those who can appreciate that sort of beauty. — E. T. BELL, Review of “Contributions to the History of Determinants, 1900–1920”, by Sir Thomas Muir (1931)
Like the matrix inverse, the determinant is a quantity that rarely needs to be computed. The common fallacy that the determinant is a measure of ill conditioning is displayed by the observation that if is orthogonal then det(aQ) = andet(Q ) = ±an , which can be made arbitrarily small or large despite the fact that aQ is perfectly conditioned. Of course, we could normalize the matrix before taking its determinant and define, for example,
where D –1 A has rows of unit 2-norm. This function is called the Hadamard condition number by Birkhoff [99, 1975], because Hadamard’s determinantal inequality (see Problem 13.11) implies that (A) > 1, with equality if and only if A has mutually orthogonal rows. Unless A is already row equilibrated (see Problem 13.13), (A) does not relate to the conditioning of linear systems in any straightforward way.
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(13.34)
where r is the number of row interchanges during the elimination. If we use (13.34) then, barring underflow and overflow, the computed determinant = fl(det(A)) satisfies where |θ n| < γn , so we have a tiny relative perturbation of the exact determinant of a perturbed matrix A + ∆A, where ∆A is bounded in Theorem 9.3. In other words, we have almost the right determinant for a slightly perturbed matrix (assuming the growth factor is small). However, underflow and overflow in computing the determinant are quite likely, and the determinant itself may not be a representable number. One possibility is to compute log |det(A ) log |uii |; as pointed out by Forsythe and Moler [396, 1967, p. 55], however, the computed sum may be inaccurate due to cancellation. Another approach, used in LINPACK, is to compute and represent det(A) in the form y × 10e, where 1 < |y| < 10 and e is an integer exponent. 13.5.1. Hyman’s Method In §1.16 we analysed the evaluation of the determinant of a Hessenberg matrix H by GE. Another method for computing det(H) is Hyman’s method [594, 1957 ], which is superficially similar to GE but algorithmically different. Hyman’s method has an easy derivation in terms of LU factorization that leads to a very short error analysis. Let bean unreduced upper Hessenberg matrix (h i+ 1 , i 0 for all i) and write
The matrix H 1 is H with the first row cyclically permuted to the bottom, so det(H 1 ) = (–1) n -1 det(H). Since T is a nonsingular upper triangular matrix, we have the LU factorization (13.35) from which we obtain det(H 1) = det(T )(η – hTT– 1 y ). Therefore det(H) = (–1) n -1
det(T)(η – hTT - 1y).
(13.36)
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Hyman's method consists of evaluating (13.36) in the natural way: by solving the triangular system Tx = y by back substitution, then forming η – hTx and its product with det(T ) = h 2 1 h 32 . . . hn,n- 1 . For the error analysis we assume that no underflows or overflows occur. From the backward error analysis for substitution (Theorem 8.5) and for an inner product ((3.4)) we have immediately, on defining µ := η – hTT– 1 y ,
where Since fl(det(T)) = det(T)(1 + δ1). . .(1 + δn -1), |δ i | < u, the computed determinant satisfies
where |θ n| < γn . This is not a backward error result, because only one of the two Ts in this expression is perturbed. However, we can write det( T ) = det(T + ∆T)(1 + θ( n -1)2), so that
We conclude that the computed determinant is the exact determinant of a perturbed Hessenberg matrix in which each element has undergone a relative perturbation not exceeding γn2 - n + 1 n 2 u , which is a very satisfactory result. In fact, the constant γn 2 - n+1 may be reduced to γ2 n -1 by a slightly modified analysis; see Problem 13.14.
13.6. Notes and References Classic references on matrix inversion are Wilkinson’s paper Error Analysis of Direct Methods of Matrix Inversion [1085, 1961] and his book Rounding Errors in Algebraic Processes [1088, 1963]. In discussing Method 1 of §13.2.1, Wilkinson says “The residual of X as a left-hand inverse may be larger than the residual as a right-hand inverse by a factor as great as ||L|| ||L - 1 || . . . We are asserting that the computed X is almost invariably of such a nature that XL – I is equally small” [1088, 1963, p. 107]. Our experience concurs with the latter statement. Triangular matrix inversion provides a good example of the value of rounding error analysis: it helps us identify potential instabilities, even if they are rarely manifested in practice, and it shows us what kind of stability is guaranteed to hold. In [1085, 1961] Wilkinson identifies various circumstances in which triangular matrix inverses are obtained to much higher accuracy than the bounds
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of this chapter suggest. The results of §8.2 provide some insight. For example, if T is a triangular M-matrix then, as noted after Corollary 8.10, its inverse is computed to high relative accuracy, no matter how ill conditioned L may be. Sections 13.2 and 13.3 are based closely on Du Croz and Higham [322, 1992 ]. Method D in §13.3.4 is used by the Hewlett-Packard HP-15C calculator, for which the method’s lack of need for extra storage is an important property [523, 1982]. Higham [560, 1995] gives error analysis of a divide-and-conquer method for inverting a triangular matrix that has some attractions for parallel computation. GJE is an old method. It was discovered independently by the geodesist Wilhelm Jordan (1842-1899) (not the mathematician Camille Jordan (18381922)) and B.-I. Clasen [12, 1987]. An Algol routine for inverting positive definite matrices by GJE was published in the Handbook for Automatic Computation by Bauer and Reinsch [83, 1971]. As a means of solving a single linear system, GJE is 50% more expensive than GE when cost is measured in flops; the reason is that GJE takes O ( n 3 ) flops to solve the upper triangular system that GE solves in n 2 flops. However, GJE has attracted interest for vector computing because it maintains full vector lengths throughout the algorithm. Hoffmann [577, 1987] found that it was faster than GE on the CDC Cyber 205, a now-defunct machine with a relatively large vector startup overhead. Turing [1027, 1948] gives a simplified analysis of the propagation of errors in GJE, obtaining a forward error bound proportional to κ(A). Bauer [80, 1966] does a componentwise forward error analysis of GJE for matrix inversion and obtains a relative error bound proportional to for symmetric positive definite A. Bauer’s paper is in German and his analysis is not easy to follow. A summary of Bauer’s analysis (in English) is given by Meinguet [746, 19 6 9 ]. The first thorough analysis of the stability of GJE was by Peters and Wilkinson [828, 1975]. Their paper is a paragon of rounding error analysis. Peters and Wilkinson observe the connection with GE, then devote their attention to the “second stage” of GJE, in which Ux = y is solved. They show that each component of x is obtained by solving a lower triangular system, and they deduce that, for each i, (U + ∆U i )x( i ) = y + ∆y i , where |∆Ui | < γn|U| and |∆yi | < γn |y|, and where the ith component of x( i ) is the ith component of They then show that has relative error bounded by but that does not necessarily have a small backward error. The more direct approach used in our analysis is similar to that of Dekker and Hoffmann [276, 1989], who give a normwise analysis of a variant of GJE that uses row pivoting (elimination across rows) and column interchanges. Our componentwise
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bounds (13.29)–(13.32) are new. Goodnight [473, 1979] describes the use of GJE in statistics for solving least squares problems. Error analysis of Hyman’s method is given by Wilkinson [1084, 1960], [1088, 196 3, pp. 147-154], [1089, 196 5, pp. 426-431]. Although it dates from the 1950s, Hyman’s method is not obsolete: it has found use in methods designed for high-performance computers; see Ward [1065, 1976], Li and Zeng [701, 1992], and Dubrulle and Golub [324, 1994].
13.6.1. LAPACK Routine xGETRI computes the inverse of a general square matrix by Method B using an LU factorization computed by xGETRF . The corresponding routine for a symmetric positive definite matrix is xPOTRI , which uses Method D, with a Cholesky factorization computed by xPOTRF . Inversion of a symmetric indefinite matrix is done by xSYTRI . Triangular matrix inversion is done by xTRTRI , which uses Method 2C. None of the LAPACK routines compute the determinant, but it is easy for the user to evaluate it after computing an LU factorization.
Problems 13.1. Reflect on this cautionary tale told by Acton [4,
1970,
p. 246].
“It was 1949 in Southern California. Our computer was a very new CPC (model 1, number 1) —a 1-second-per-arithmetic-operation clunker that was holding the computational fort while an early electronic monster was being coaxed to life in an adjacent room, From a nearby aircraft company there arrived one day a 16 × 16 matrix of 10-digit numbers whose inverse was desired . . . We labored for two days and, after the usual number of glitches that accompany any strange procedure involving repeated handling of intermediate decks of data cards, we were possessed of an inverse matrix. During the checking operations . . . it was noted that, to eight significant figures, the inverse was the transpose of the original matrix! A hurried visit to the aircraft company to explore the source of the matrix revealed that each element had been laboriously hand computed from some rather simple combinations of sines and cosines of a common angle. It took about 10 minutes to prove that the matrix was, indeed, orthogonal!” 13.2. Rework the analysis of the methods of §13.2.2 using the assumptions (12.3) and (12.4), thus catering for possible use of fast matrix multiplication techniques.
MATRIX INVERSION
286 13.3. Show that for any nonsingular matrix A,
This inequality shows that the left and right residuals of X as an approximation to A–1 can differ greatly only if A is ill conditioned. 13.4. (Mendelssohn [748, 1956]) Find parametrized 2 × 2 matrices A and X such that the ratio ||AX – I||/||XA – I|| can be made arbitrarily large. 13.5. Let X and Y be approximate inverses of
that satisfy
and Show that
Derive forward error bounds for
and
Interpret all these bounds.
13.6. What is the relation between the matrix on the front cover of the LAPACK Users’ Guide [17, 1995] and that on the back cover? Answer the same question for the LINPACK Users’ Guide [307, 1979]. 13.7. Show that for any matrix having a row or column of 1s, the elements of the inverse sum to 1. 13.8. Let X = A + iB be nonsingular. Show that X-1 expressed in terms of the inverse of the real matrix of order 2 n,
can
be
Show that if X is Hermitian positive definite then Y is symmetric positive definite. Compare the economics of real versus complex inversion. 13.9. For a given nonsingular and X A -1, it is interesting to ask what is the smallest ε such that X + ∆X = (A + ∆A)-1 with ||∆ X|| < and We require (A + ∆A)(X + ∆X) = I, or A ∆X + ∆AX + ∆A∆X = I – AX. It is reasonable to drop the second-order term, leaving a generalized Sylvester equation that can be solved using singular value decompositions of A and X (cf. §15.2). Investigate this approach computationally for a variety of A and methods of inversion.
PROBLEMS
287
13.10. For a nonsingular and given integers i and j, under what conditions is det(A) independent of aij ? What does this imply about the suitability of det(A ) for measuring conditioning? 13.11. Prove Hadamard’s inequality for
where ak = A(:, k). When is there equality? (Hint: use the QR factorization.) 13.12. (a) Show that if AT = QR is a QR factorization then the Hadamard condition number where pi = ||R(:, i)||2. (b) Evaluate for A = U(1) (see (8.2)) and for the Pei matrix A = (a – 1)I + eeT. 13.13. (Guggenheimer, Edelman, and Johnson [486, a nonsingular
1995])
(a) Prove that for
(Hint: apply the arithmetic-geometric mean inequality to the n numbers where the σ i are the singular values of A.) (b) Deduce that if A has rows of unit 2-norm then
where
is the Hadamard condition number.
13.14. Show that Hyman’s method for computing det(H), where is an unreduced upper Hessenberg matrix, computes the exact determinant of H + ∆H where |∆ H| < γ2 n - 1 |H|, barring underflow and overflow. What is the effect on the error bound of a diagonal similarity transformation H D - 1 HD, where D = diag(di ), di 0? 13.15. What is the condition number of the determinant? 13.16. (RESEARCH PROBLEM) Obtain backward and forward error bounds for GJE applied to a diagonally dominant matrix. Peters and Wilkinson [828, 1975] state that “it is well known that Gauss–Jordan is stable” for a diagonally dominant matrix, but a proof does not seem to have been published.
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Chapter 14
Condition Number Estimation
Most of LAPACK’S condition numbers and error bounds are based on estimated condition numbers . . . The price one pays for using an estimated rather than an exact condition number is occasional (but very rare) underestimates of the true error; years of experience attest to the reliability of our estimators, although examples where they badly underestimate the error can be constructed. — E. ANDERSON et al,, LAPACK Users’ Guide, Release 2.0 (1995)
The importance of the counter-examples is that they make clear that any effort toward proving that the algorithms always produce useful estimations is fruitless. It may be possible to prove that the algorithms produce useful estimations in certain situations, however, and this should be pursued. An effort simply to construct more complex algorithms is dangerous. — A. K. CLINE and R. K. REW, A Set of Counter-Examples to Three Condition Number Estimators (1983)
Singularity is almost invariably a clue. — SIR ARTHUR CONAN DOYLE, The Boscombe Valley Mystery “(1892)
289
Next
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CONDITION NUMBER ESTIMATION
14.1. How to Estimate Componentwise Condition Numbers When bounding the forward error of a computed solution to a linear system we would like to obtain the bound with an order of magnitude less work than is required to compute the solution. For a dense n × n system, where the solution process usually requires O(n 3) operations, we need to compute the bound in O(n 2) operations. An estimate of the bound that is correct to within a factor 10 is usually acceptable, because it is the magnitude of the error that is of interest, not its precise value. In the perturbation theory of Chapter 7 for linear equations we obtained perturbation bounds that involve the condition numbers
and their variants. To compute these condition numbers exactly we need to compute A–1 , which requires O(n 3 ) operations, even if we are given a factorization of A. Is it possible to produce reliable estimates of both condition numbers in O(n 2) operations? The answer is yes, but to see why we first need to rewrite the expression for condE , f(A, x). Consider the general expression where d is a given nonnegative vector (thus, d = f + E|x| for cond E , f (A, x)); note that the practical error bound (7.27) is of this form. T Writing D = diag(d) and e = [1, 1, . . . . 1] , we have (14.1) Hence the problem is equivalent to that of estimating If we can estimate this quantity then we can estimate any of the condition numbers or perturbation bounds for a linear system. There is an algorithm, described in §14.3, that produces reliable order-of-magnitude estimates of ||B||1, for an arbitrary B, by computing just a few matrix-vector products Bx and BTy for carefully selected vectors x and y and then approximating ||B|| 1 ||Bx|| 1 /||x||1. If we assume that we have a factorization of A (say, PA = LU or A = QR), then we can form the required matrix-vector products for B = A- 1 D in O(n 2) flops. Since ||B||1 = it follows that we can use the algorithm to estimate in O(n 2) flops. Before presenting the l-norm condition estimation algorithm, we describe a more general method that estimates ||B||p for any This more general method is of interest in its own right and provides insight into the special case p = 1.
14.2 THE p-NORM
POWER
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METHOD
14.2. The p-Norm Power Method An iterative “power method” for computing ||A||p was proposed by Boyd in 1974 [139, 1974]. When p = 2 it reduces to the usual power method applied to ATA. We use the notation dualp (x) to denote any vector y of unit qnorm such that equality holds in the Holder inequality (this normalization is different from the one we used in (6.3), but is more convenient for our present purposes). Throughout this section, p > 1 and q is defined by p -1 + q-1 = 1. Algorithm 14.1 (p-norm power method). Given and this algorithm computes γ and x such that γ < ||A||p and ||Ax||p = γ||x||p . x = x0 repeat y z if
/||x0 ||p
= Ax = AT dualp (y) ||z||q < zTx γ = ||y||p quit end x = dualq (z)
end Cost: 4rmn flops (for r iterations). There are several ways to derive Algorithm 14.1. Perhaps the most natural way is to examine the optimality conditions for maximization of
First, we note that the subdifferential (that is, the set of subgradients) of an arbitrary vector norm ||·|| is given by [378, 1987, p. 379]
If x 0 then x 0,
from the Holder inequality, and so, if
It can also be shown that if A has full rank, (14.2)
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CONDITION NUMBER ESTIMATION
We assume now that A has full rank, 1 < p < and x is easy to see that there is a unique vector dualp ( x ), so element, that is, ||x||p is differentiable. Hence we have
0. Then it has just one
(14.3) The first-order Kuhn–Tucker condition for a local maximum of F is therefore
Since dualq (dual p (x)) = x/||x||p if p
1,
this equation can be written (14.4)
The power method is simply functional iteration applied to this transformed set of Kuhn–Tucker equations (the scale factor is irrelevant since F(ax) = F(x)). For the 1- and -norms, which are not everywhere differentiable, a different derivation can be given. The problem can be phrased as one of maximizing the convex function F(x) := ||Ax||p over the convex set S := {x : ||x||p < 1}. The convexity of F and S ensures that, for any at least one vector g exists such that (14.5) Vectors g satisfying (14.5) are called subgradients of F (see, for example, [378, 198 7, p. 364]). Inequality (14.5) suggests the strategy of choosing a subgradient g and moving from u to a point that maximizes gT (υ – u), T that is, a vector that maximizes g υ. Clearly, υ * = dualq ( g). Since, from (14.2), g has the form AT dualp (Ax), this completes the derivation of the iteration. For all values of p the power method has the desirable property of generating an increasing sequence of norm approximations. Lemma 14.2. In Algorithm 14.1, the vectors from the kth iteration satisfy
The first inequality in (ii) is strict if convergence is not obtained on the kth iteration.
14.2 THE p-NORM Proof.
POWER
293
METHOD Then
For the last part, note that, in view of (i), the convergence test “||zk||q < can be written as “||zk||q < ||yk||p ” . It is clear from Lemma 14.2 (or directly from the Holder inequality) that the convergence test “||z|| q < z T x” in Algorithm 14.1 is equivalent to “||z||q = z T x” and, since ||x||p = 1, this is equivalent to x = dualq ( z ). Thus, although the convergence test compares two scalars, it is actually testing for equality in the vector equation (14.4). The convergence properties of Algorithm 14.1 merit a careful description. First, in view of Lemma 14.2, the scalars γk = ||yk||p form an increasing and convergent sequence. This does not necessarily imply that Algorithm 14.1 converges, since the algorithm tests for convergence of the xk , and these vectors could fail to converge. However, a subsequence of the xk must converge to a limit, say. Boyd [139, 1974] shows that if is a strong local maximum of F with no zero components, then linearly. If Algorithm 14.1 converges it converges to a stationary point of F( x ) when 1 < p < Thus, instead of the desired global maximum ||A||p , we may obtain only a local maximum or even a saddle point. When p = 1 or if the algorithm converges to a point at which F is not differentiable, that point need not even be a stationary point. On the other hand, for p = 1 or Algorithm 14.1 terminates in at most n + 1 iterations (assuming that when dualp or dualq is not unique an extreme point of the unit ball is taken), since the algorithm moves between the vertices ei of the unit ball in the 1norm, increasing F on each stage (x = ±ei for p = 1, and dual p (y) = ±ei for P = An example where n iterations are required for p = 1 is given in Problem 14.2. For two special types of matrix, more can be said about Algorithm 14.1. (1) If A = xyT (rank 1), the algorithm converges on the second step with γ = ||A||p = ||x||p||y||q , whatever x0 . (2) Boyd [139, 1974] shows that if A has nonnegative elements, ATA is irreducible, 1 < p < and x0 has positive elements, then the xk converge and γ k ||A||p . For values of p strictly between 1 and the convergence behaviour of Algorithm 14.1 is typical of a linearly convergent method: exact convergence is not usually obtained in a finite number of steps and arbitrarily many steps can
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be required for convergence, as is well-known for the 2-norm power method. Fortunately, there is a method for choosing a good starting vector that can be combined with Algorithm 14.1 to produce a reliable norm estimator; see the Notes and References and Problem 14.1. We now turn our attention to the extreme values of p: 1 and
14.3. LAPACK l-Norm Estimator Algorithm 14.1 has two remarkable properties when p = 1: it almost always converges within four iterations (when x0 = [1, 1, . . . , 1]T, say) and it frequently yields ||A||1 exactly. This rapid, finite termination is also obtained for p = and is related to the fact that Algorithm 14.1 moves among the finite set of extreme points of the unit ball. Numerical experiments suggest that the accuracy is about the same for both norms but that slightly more iterations are required on average for p = Hence we will confine our attention to the 1-norm. The l-norm version of Algorithm 14.1 was derived independently of the general algorithm by Hager [492, 1984] and can be expressed as follows. The notation ξ = sign(y) means that ξ i = 1 or –1 according as yi > 0 or yi < 0. We now specialize to square matrices. Algorithm 14.3 (1-norm power method). Given computes γ and x such that γ < ||A||1 and ||Ax||1 = x = n– 1e repeat y = Ax ξ = sign(y) z = A Tξ if γ = ||y||1 quit end x = e j, where |zj| = end
γ||x|| 1 .
this algorithm
(smallest such j)
Numerical experiments show that the estimates produced by Algorithm 14.3 are frequently exact (γ = ||A|| 1), usually “acceptable” (γ > ||A||1/10), and sometimes poor (γ < ||A||1/10). An important question for any norm or condition estimator is whether there exists a “counterexample”—a parametrized matrix for which the quotient “estimate divided by true norm” can be made arbitrarily small (or large, depending on whether the estimator produces a lower bound or an upper
14.3 LAPACK 1-N ORM E STIMATOR
295
bound) by varying a parameter. A general class of counterexample for Algorithm 14.3 is given by the matrices
where Ce = CTe = 0 (there are many possible choices for C ). For any such matrix, Algorithm 14.3 computes y = n – 1e, ξ = e, z = e, and hence the algorithm terminates at the end of the first iteration with
The problem is that the algorithm stops at a local maximum that can differ from the global one by an arbitrarily large factor. A more reliable and more robust algorithm is produced by the following modifications of Higham [537, 1988]. Definition of estimate. To overcome most of the poor estimates, γ is redefined as
where
The vector b is considered likely to “pick out” any large elements of A in those cases where such elements fail to propagate through to y. Convergence test. The algorithm is limited to a minimum of two and a maximum of five iterations. Further, convergence is declared after computing ξ if the new ξ is the same as the previous one; this event signals that convergence will be obtained on the current iteration and that the next (and final) multiplication ATξ is unnecessary. Convergence is also declared if the new ||y||1 is no larger than the previous one. This nonincrease of the norm can happen only in finite precision arithmetic and signals the possibility of a vertex ej being revisited—the onset of “cycling.” The improved algorithm is as follows. This algorithm is the basis of all the condition number estimation in LAPACK. Algorithm 14.4 (LAPACK norm estimator). Given this algorithm computes γ and υ = Aw such that γ < ||A|| 1 with ||υ||1 /||w|| 1 = γ (w is not returned). υ = A(n – le) if n = 1, quit with γ = |υ 1|, end γ = ||υ| |1 ξ = sign(υ)
296
CONDITION NUMBER ESTIMATION x = A Tξ k=2 repeat j = min{i: |xi | = υ = Aej γ = ||υ| |1 if sign(υ) = ξ or γ < ξ = sign(υ) x = A Tξ k = k+ 1
}
goto (*), end
until i+ 1 (*) xi = (–1) x = Ax i f 2||x||1 /(3n) > γ then υ = x γ = 2||x||1 /(3n) end Algorithm 14.4 can still be “defeated”: it returns an estimate 1 for matrices A(θ ) of the form A(θ) = I + θP, where P = PT, Pe = 0, Pe1 = 0, Pb = 0.
(14.6)
(P can be constructed as I – Q where Q is the orthogonal projection onto span{e, e1, b}.) Indeed, the existence of counterexamples is intuitively obvious since Algorithm 14.4 samples the behaviour of A on less than n vectors in Numerical counterexamples (not parametrized) can be constructed automatically by direct search, as described in §24.3.1. Despite these weaknesses, practical experience with Algorithm 14.4 shows that it is very rare for the estimate to be more than three times smaller than the actual norm, independent of the dimension n. Therefore Algorithm 14.4 is, in practice, a very reliable norm estimator. The number of matrix-vector products required is at least 4 and at most 11, and averages between 4 and 5. There is an analogue of Algorithm 14.3 for complex matrices, in which ξi is defined as yi /|yi | if yi 0 and 1 otherwise. In the corresponding version of Algorithm 14.4 the test for repeated ξ vectors is removed, because ξ now has noninteger, complex components and so is unlikely to repeat. It is interesting to look at a performance profile of Algorithm 14.4. A performance profile is a plot of some measure of the performance of an algorithm versus a problem parameter. In this case, the natural measure of performance is the underestimation ratio, γ/||A|| 1. Figure 14.1 shows the performance profile for a 5 × 5 matrix A( θ) of the form (14.6), with P constructed as
14.4 O THER C ONDITION E STIMATORS
297
Figure 14.1. Underestimation ratio for Algorithm 14.4 for 5 × 5 matrix A(O) of (14.6) with 150 equally spaced values of [0,10]. described above (because of rounding errors in constructing A(θ ) and within the algorithm, the computed norm estimates differ from those that would be produced inexact arithmetic). The jagged nature of the performance curve is typical for algorithms that contain logical tests and branches. Small changes in the parameter θ, which themselves result in different rounding errors, can cause the algorithm to visit different vertices in this example.
14.4. Other Condition Estimators The first condition estimator to be widely used is the one employed in LINPACK. It was developed by Cline, Moler, Stewart, and Wilkinson [216, 1979]. The idea behind this condition estimator originates with Gragg and Stewart [476, 1976], who were interested in computing an approximate null vector rather than estimating the condition number itself. We will describe the algorithm as it applies to a triangular matrix T There are three steps: 1. Choose a vector d such that ||y|| is as large as possible relative to ||d||, where TTy = d. 2. Solve Tx = y.
CONDITION NUMBER ESTIMATION
298 3. Estimate
In LINPACK the norm is the l-norm, but the algorithm can also be used for the 2-norm or the cm-norm. The motivation for step 2 is based on a singular value decomposition analysis. Roughly, if ||y||/||d|| is large then will almost certainly be at least as large, and it could be a much better estimate. Notice that TTTx = d, so the algorithm is related to the power method on the matrix (TTT) –1 with the specially chosen starting vector d. To examine step 1 more closely, suppose that T = UT is lower triangular and note that the equation Uy = d can be solved by the following columnoriented (saxpy) form of substitution:
end The idea is to choose the elements of the right-hand side vector d adaptively as the solution proceeds, with d j = ±1. At the jth stage of the algorithm d . . . , dj+1 have been chosen and yn , . . . , yj+1 are known. The next element dj {+1, –1} is chosen so as to maximize a weighted sum of dj – pj and the partial sums p1, . . . , pj, which would be computed during the next execution of statement (*) above. Hence the algorithm looks ahead, trying to gauge the effect of the choice of d j on future solution components. This heuristic algorithm for choosing d is expressed in detail as follows. Algorithm 14.5 (LINPACK condition estimator). Given a nonsingular upper triangular matrix and a set of nonnegative weights {w i }, this algorithm computes a vector y such that Uy = d, where the elements dj = ±1 are chosen to make ||y|| large.
14.4 O THER C ONDITION E STIMATORS
299
p(1:j– 1 ) = p -(1:j–1) end end cost: 4n 2 flops. LINPACK takes the weights wj 1, though another possible (but more expensive) choice would be wj = 1/|ujj|, which corresponds to how pj is weighted in the expression yj = (dj – pj)/ujj. To estimate ||A- 1 || for a full matrix A, the LINPACK estimator makes use of an LU factorization of A. Given PA = LU, the equations solved are UTz = d, LTy = z, and AX = PTy, where for the first system d is constructed by the analogue of Algorithm 14.5 for lower triangular matrices; the estimate is ||x||1 /||y||1 ||A - 1 ||1. Since d is chosen without reference to L, there is an underlying assumption that any ill condition in A is reflected in U. This assumption may not be true; see Problem 14.3. In contrast to the LAPACK norm estimator, the LINPACK estimator requires explicit access to the elements of the matrix. Hence the estimator cannot be used to estimate componentwise condition numbers. Furthermore, separate code has to be written for each different type of matrix and factorization. Consequently, while LAPACK has just a single norm estimation routine, which is called by many other routines, LINPACK has multiple versions of its algorithm, each tailored to the specific matrix or factorization. Several years after the LINPACK condition estimator was developed, several parametrized counterexamples were found by Cline and Rew [217, 1983]. Numerical counterexamples can also be constructed by direct search, as shown in §24.3.1. Despite the existence of these counterexamples the LINPACK estimator has been widely used and is regarded as being almost certain to produce an estimate correct to within a factor 10 in practice. A 2-norm condition estimator was developed by Cline, Corm, and Van Loan [218, 1982, Algorithm 1]; see also Van Loan [1043, 1987] for another explanation. The algorithm builds on the ideas underlying the LINPACK estimator by using “look-behind” as well as look-ahead. It estimates σ min (R) = or σm a x (R) = ||R||2 for a triangular matrix R, where σ min and σ m a x denote the smallest and largest singular values, respectively. Full matrices can be treated if a factorization A = QR is available ( Q orthogonal, R upper triangular), since R and A have the same singular values. The estimator performs extremely well in numerical tests, often producing an estimate that has some correct digits [218, 1982], [534, 1987]. No counterexamples to the estimator were known until Bischof [103, 1990] obtained counterexamples as a by-product of the analysis of a different but related method, mentioned at the end of this section. All the methods described so far have the property that when applied repeatedly to a given matrix they always produce the same estimate. Another
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C ONDITION N UMBER E STIMATION
approach is to introduce some randomness, so that the output of the method depends on the particular random numbers chosen. A natural idea along these lines is to apply the power method to the matrix (AAT) -1 with a randomly chosen starting vector. If a factorization of A is available, the power method vectors can be computed inexpensively by solving linear systems with A and AT. Analysis based on the singular value decomposition suggests that there is a high probability that a good estimate of ||A - 1 ||2 will be obtained. This notion is made precise by Dixon [306, 1983], who proves the following result. Theorem 14.6 (Dixon). Let be nonsingular and let θ > 1 be a constant. If is a random vector from the uniform distribution on the unit sphere Sn = then the inequality (14.7) holds with probability at least 1 – 0.8θ - k / 2 n 1/2
(k > 1).
Note that the left-hand inequality in (14.7) always holds; it is only the right-hand inequality that is in question. For k = 1, (14.7) can be written as
which suggests the simple estimate where x is chosen randomly from the uniform distribution on Sn. Such vectors x can be generated from the formula where z1, . . . , zn are independent random variables from the normal N(0, 1) distribution [668, 1981, p. 130]. If, for example, n = 100 and θ has the rather large value 6400 then inequality (14.7) holds with probability at least 0.9. In order to take a smaller constant θ, for fixed n and a desired probability, we can use larger values of k. If k = 2j is even then we can simplify (14.7), obtaining (14.8) and the minimum probability stated by the theorem is 1 - 0.8θ - j n 1/2. Taking j = 3, for the same value n = 100 as before, we find that (14.8) holds with probability at least 0.9 for the considerably smaller value θ = 4.31. Probabilistic condition estimation has not yet been adopted in any major software packages, perhaps because the other techniques work so well. For more on the probabilistic power method approach see Dixon [306, 1983], Higham [534, 1987], and Kuczynski and Wozniakowski [676, 1992] (who also analyse the more powerful Lanczos method with a random starting vector). For a probabilistic condition estimation method of very general applicability
14.5 C ONDITION N UMBERS
OF
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301
see Kenney and Laub [652, 1994] and Gudmundsson, Kenney, and Laub [485, 1995 ]. The condition estimators described above assume that a single estimate is required for a matrix given in its entirety. Condition estimators have also been developed for more specialized situations. Bischof [103, 1990] develops a method for estimating the smallest singular value of a triangular matrix which processes the matrix a row or a column at a time. This “incremental condition estimation” method can be used to monitor the condition of a triangular matrix as it is generated, and so is useful in the context of matrix factorization such as the QR factorization with column pivoting. The estimator is generalized to sparse matrices by Bischof, Lewis, and Pierce [104, 1990]. Barlow and Vemulapati [67, 1992] develop a l-norm incremental condition estimator with look-ahead for sparse matrices. Condition estimates are also required in applications where a matrix factorization is repeatedly updated as a matrix undergoes low rank changes. Algorithms designed for a recursive least squares problem and employing the Lanczos method are described by Ferng, Golub, and Plemmons [372, 1991 ]. Pierce and Plemmons [831, 1992 ] describe an algorithm for use with the Cholesky factorization as the factorization is updated, while Shroff and Bischof [918, 1992] treat the QR factorization.
14.5. Condition Numbers of Tridiagonal Matrices For a bidiagonal matrix B, |B–1| = M(B)–1 (see §8.3), so the condition numbers κ E,f and condE,f can be computed exactly with an order of magnitude less work than is required to compute B-1 explicitly. This property holds more generally for several types of tridiagonal matrix, as a consequence of the following result. Recall that the LU factors of a tridiagonal matrix are bidiagonal and may be computed using the formulae (9.16). Theorem 14.7. If the nonsingular tridiagonal matrix has the LU factorization A = LU and |L||U| = |A|, then |U - 1 |L- 1 | = |A- 1|. Proof. Using the notation of (9.15), |L||U| = |A| = |LU| if and only if, for all 2, that is, if (14.9) Using the formulae
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CONDITION NUMBER ESTIMATION
we have
Thus, in view of (14.9), it is clear that |U - 1 L - 1|ij = (|U- 1 ||L - 1 |) z j, as required. Since L and U are bidiagonal, |U- 1 | = M(U)-1 and |L- 1 | = M(L)- 1 . Hence, if |A| = |L||U|, then, from Theorem 14.7, (14.10) It follows that we can compute any of the condition numbers or forward error bounds of interest exactly by solving two bidiagonal systems. The cost is O(n) flops, as opposed to the O(n2) flops needed to compute the inverse of a tridiagonal matrix. When does the condition |A| = |L||U| hold? Theorem 9.11 shows that it holds if the tridiagonal matrix A is symmetric positive definite, totally positive, or an M-matrix. So for these types of matrix we have a very satisfactory way to compute the condition number. If A is tridiagonal and diagonally dominant by rows, then we can compute in O(n) flops an upper bound for the condition number that is not more than a factor 2n – 1 too large.
14.5 C ONDITION N UMBERS
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303
Theorem 14.8. Suppose the nonsingular, row diagonally dominant tridiagonal matrix has the LU factorization A = LU. Then, if y > 0,
Proof. We have L-1 = UA– 1 , so
where the bidiagonal matrix V = diag(u i i )-1 U has υi i |ei /ui | < 1 (see the proof of Theorem 9.12). Thus
1 and |υi , i +1| =
and the result follows on taking norms. In fact, it is possible to compute |A- 1|y exactly in O(n) operations for any tridiagonal matrix. This is a consequence of the special form of the inverse of a tridiagonal matrix. Theorem 14.9 (Ikebe). Let be tridiagonal and irreducible (that is, ai+ 1 ,i and ai,i+1 are nonzero for all i). Then there are vectors x, y, p, and q such that
This result says that the inverse of an irreducible tridiagonal matrix is the upper triangular part of a rank-1 matrix joined along the diagonal to the lower triangular part of another rank-1 matrix. If A is reducible then it has (or its transpose), and this blocking can be applied the block form recursively until the diagonal blocks are all irreducible, at which point the theorem can be applied to the diagonal blocks. The vectors x, y, p, and q in Theorem 14.9 can all be computed in O(n) flops, and this enables the condition numbers and forward error bounds to be computed also in O(n) flops (see Problem 14.5). Unfortunately, the vectors x, y, p, and q can have a huge dynamic range, causing the computation to break down because of overflow or underflow. For example, for the diagonally dominant tridiagonal matrix with a ii 4, ai+ 1 ,i = ai,i+ 1 1, we have (x1 = 1) |x n | θ n -1, |y1| θ -1, and |yn| θ - n , where θ = 2 + 3.73. These numerical problems can be overcome, but only at a nontrivial increase in cost. Therefore, we do not recommend the use of Theorem 14.9 for computing condition numbers. For a general tridiagonal matrix it is probably better to estimate the condition number using Algorithm 14.4.
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14.6. Notes and References The clever trick (14.1) for converting the norm into the norm of a matrix with which products are easily formed is due to Arioli, Demmel, and Duff [24, 1989]. The p-norm power method was first derived and analysed by Boyd [139, 1974] and was later investigated by Tao [993, 1984]. Tao applies the method to an arbitrary mixed subordinate norm ||A||a,β (see (6.6)), while Boyd takes the a and β-norms to be p-norms (possibly different). Algorithm 14.1 can be converted to estimate ||A||a,β by making straightforward modifications to the norm-dependent terms. An algorithm that estimates ||A||p using the power method with a specially chosen starting vector is developed by Higham [551, 1992]; the method for obtaining the starting vector is outlined in Problem 14.1. The estimate produced by this algorithm is always within a factor n 1 – 1 /p of ||A||p and the cost is about 70n 2 flops. A MATLAB M-file pnorm implementing this method is part of the Test Matrix Toolbox (see Appendix E). The finite convergence of the power method for p = 1 and p = 00 holds more generally: if the power method is applied to the norm ||·|| a,β and one of the a and β norms is polyhedral (that is, its unit ball has a finite number of extreme points), then the iteration converges in a finite number of steps. Moreover, under a reasonable assumption, this number of steps can be bounded in terms of the number of extreme points of the unit balls in the a-norm and the dual of the β-norm. See Bartels [75, 1991] and Tao [993, 1984] for further details. Hager [492, 1984] gave a derivation of the 1-norm estimator based on subgradients and used the method to estimate κ 1 (A). That the method is of wide applicability y because it accesses A only through matrix-vector products was recognized by Higham, who developed Algorithm 14.4 and its complex analogue and wrote Fortran 77 implementations, which use a reverse communication interface [537, 1988], [543, 1990]. These codes are used in LAPACK, the NAG library, and various other program libraries. A version of Algorithm 14.4 dedicated to estimating κ 1 (A) is supplied with MATLAB as M-file condest. Algorithm 14.4 is also implemented in ROM on the Hewlett-Packard HP 48G and HP 48GX calculators (along with several other LAPACK routines), in a form that estimates κ 1 (A). The Hewlett-Packard implementation is instructive because it shows that condition estimation can be efficient even for small dimensions: on a standard HP 48G, inverting A and estimating its condition number (without being given a factorization of A in either case) both take about 5 seconds for n = 10, while for n = 20 inversion takes 30 seconds and condition estimation only 20 seconds. Moler [769, 1978] describes an early version of the LINPACK condition estimator and raises the question of the existence of counterexamples. An early version without look-ahead was incorporated in the Fortran code decomp
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in the book of Forsythe, Malcolm, and Moler [395, 1977]. Matrices for which condition estimators perform poorly can be very hard to find theoretically or with random testing, but for all the estimators described in this chapter they can be found quite easily by applying direct search optimization to the under- or overestimation ratio; see §24.3.1. Both LINPACK and LAPACK return estimates of the reciprocal of the condition number, in a variable rcond < 1. Overflow for a very ill conditioned matrix is thereby avoided, and rcond is simply set to zero when singularity is detected. MATLAB has a built-in function rcond that implements the LINPACK condition estimation algorithm. A simple modification to the LINPACK estimator that can produce a larger estimate is suggested by O’Leary [804, 198 0]. For sparse matrices, Grimes and Lewis [483, 1981] suggest a way to reduce the cost of the scaling strategy used in LINPACK to avoid overflow in the condition estimation. Zlatev, Wasniewski, and Schaumburg [1134, 1986] describe their experience in implementing the LINPACK condition estimation algorithm in a software package for sparse matrices. Stewart [946, 1980] describes an efficient way to generate random matrices of a given condition number and singular value distribution (see §26.3) and tests the LINPACK estimator on such random matrices. Condition estimators specialized to the (generalized) Sylvester equation have been developed by Byers [172, 1984], Kågström and Westin [624, 1989], and Kågström and Poromaa [621, 1992]. A survey of condition estimators up to 1987, which includes counterexamples and the results of extensive numerical tests, is given by Higham [534, 19 8 7 ]. Theorems 14.7 and 14.8 are from Higham [541, 1990]. That can be computed in O(n) flops for symmetric positive definite tridiagonal A was first, shown in Higham [531, 1986]. Theorem 14.9 has a long history, having been discovered independently in various forms by different authors. The earliest reference we know for the result as stated is Ikebe [600, 1979], where a more general result for Hessenberg matrices is proved. A version of Theorem 14.9 for symmetric tridiagonal matrices was proved by Bukhberger and Emel’yanenko [157, 1973]. The culmination of the many papers on inverses of tridiagonal and Hessenberg matrices is a result of Cao and Stewart on the form of the inverse of a block matrix (Aij) with Aij = 0 for i > j + s [184, 1986]; despite the generality of this result, the proof is short and elegant. Any banded matrix has an inverse with a special “low rank” structure; the earliest reference on the inverse of a general band matrix is Asplund [32, 1959]. For a recent survey on the inverses of symmetric tridiagonal and block tridiagonal matrices see Meurant [751, 1992 ]. For symmetric positive definite tridiagonal A the standard way to solve
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AX = b is by using a Cholesky or LDLT factorization, rather than an LU factorization. The LINPACK routine SPTSL uses a nonstandard “LUB” factorization resulting from the BABE (“burn at both ends”) algorithm, which eliminates from the middle of the matrix to the top and bottom simultaneously (see the LINPACK Users’ Guide [307, 1979, Chap. 7] and Higham [531, 19 86] ). The results of §14.5 are applicable to all these factorization, with minor modifications.
14.6.1. LAPACK Algorithm 14.4 is implemented in routine xLACON , which has a reverse communication interface. The LAPACK routines xPTCON and xPTRFS for symmetric positive definite tridiagonal matrices compute condition numbers using (14.10); the LAPACK routines xGTCON and xGTRFS for general tridiagonal matrices use Algorithm 14.4. LINPACK’S tridiagonal matrix routines do not incorporate condition estimation.
Problems 14.1. (Higham [551, 1992]) The purpose of this problem is to develop a noniterative method for choosing a starting vector for Algorithm 14.1. The idea is to choose the components of x in the order x1, x2, . . . . xn in an attempt to maximize ||Ax||p/||x||p. Suppose x1, . . . . xk-1 satisfying ||x(1:k-1)||p = 1 have been determined and let γk -1 = ||A( : , 1 :k-1)x(1:k-1)||p. We now try to choose xk, and at the same time revise x(1:k –1), to give the next partial product a larger norm. Defining
we set
where
Then ||x(1:k)||p = 1 and
Develop this outline into a practical algorithm. What can you prove about the quality of the estimate ||Ax||p /||x||p that it produces?
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PROBLEMS 14.2. (Higham [543, (tij ) be defined by
1990])
Let the n × n symmetric tridiagonal matrix Tn ( a ) =
For example, T6 (a) is given by
Note that, for all a, ||Tn (a)e n- 1||1 = ||Tn (a)||1. Show that if Algorithm 14.3 is applied to Tn (a) with 0 < a < 1 then x = ei -1 on the ith iteration, for i = 2, . . ., n, with convergence on the nth iteration. Algorithm 14.4, however, terminates after five iterations with y5 = Tn (a)e4, and
Show that the extra estimate saves the day, so that Algorithm 14.4 returns a final estimate that is within a factor 3 of the true norm, for any a < 1. 14.3. Let PA = LU be an LU factorization with partial pivoting of A 1 Show that
14.4. (Higham [537, 1988]) Investigate the behaviour of Algorithms 14.3 and 14.4 for the Pei matrix, A = aI + eeT ( a > 0), and for the upper bidiagonal matrix with 1s on the diagonal and the first superdiagonal. 14.5. (Ikebe [600, 1979], Higham [531, 1986]) Let be nonsingular, tridiagonal, and irreducible. By equating the last columns in AA -1 = I and the first rows in A –1 A = I, show how to compute the vectors x and y in Theorem 14.9 in O(n) flops. Hence obtain an O(n) flops algorithm for computing where d > 0.
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14.6. The representation of Theorem 14.9 for the inverse of nonsingular, tridiagonal, and irreducible involves 472 parameters, yet A depends only on 3n – 2 parameters. Obtain an alternative representation that involves only 3n – 2 parameters. (Hint: symmetrize the matrix.) 14.7. (R ESEARCH P ROBLEM ) (Demmel [286, 1992 ]) Show that estimating ||A - 1|| to within a factor depending only on the dimension of A is at least as expensive as computing A– 1 . 14.8. (RESEARCH PROBLEM) Let be diagonally dominant by rows, let A = LU be an LU factorization, and let y > 0. What is the maximum size of This is an open problem raised in [541, 1990]. In a small number of numerical experiments with full random matrices the ratio has been found to be less than 2 [541, 1990], [790, 1986].
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Chapter 15
The Sylvester Equation
We must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns, the squares corresponding to which may be termed determinants of the pth order. — J. J. SYLVESTER, Additions to the Articles, “On a New Class of Theorems, ” and “On Pasta/’s Theorem” (1850)
I have in previous papers defined a “Matrix” as a rectangular array of terms, out of which different systems of determinants may be engendered, as from the womb of a common parent; these cognate determinants being by no means isolated in their relations to one another, but subject to certain simple laws of mutual dependence and simultaneous deperition. — J. J. SYLVESTER, On the Relation Between the Minor Determinants of Linear/y Equivalent Quadratic Functions (1851)
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THE SYLVESTER EQUATION
310 The linear matrix equation AX – XB = C,
(15.1)
where and are given and is to be determined, is called the Sylvester equation. It is of pedagogical interest because it includes as special cases several important linear equation problems: 1 . linear system: Ax = c, 2 . multiple right-hand side linear system: AX = C, 3 . matrix inversion: AX = I, 4 . eigenvector corresponding to given eigenvalue b: (A – bI)x = 0, 5 . commuting matrices: AX – XA = 0. The Sylvester equation arises in its full generality in various applications. For example, the equations
show that block-diagonalizing a block triangular matrix is equivalent to solving a Sylvester equation. The Sylvester equation can also be produced from finite difference discretization of a separable elliptic boundary value problem on a rectangular domain, where A and B represent application of a difference operator in the “y” and “x” directions, respectively [935, 1991]. That (15.1) is merely a linear system is emphasized by writing it in the form (15.2) where A B := (aij B) is a Kronecker product and the vec operator stacks the columns of a matrix into one long vector. For future reference, we note the useful relation (See Horn and Johnson [581, 1991, Chap. 4] for a detailed presentation of properties of the Kronecker product and the vec operator). The mn × mn coefficient matrix in (15.2) has a very special structure, illustrated for n = 3 by
15.1 S OLVING
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In dealing with the Sylvester equation it is vital to consider this structure and not treat (15.2) as a general linear system. Since the mn eigenvalues of In A – BT Im are given by (15.3) the Sylvester equation is nonsingular precisely when A and B have no eigenvalues in common. In this chapter we briefly discuss the Schur method for solving the Sylvester equation and summarize its rounding error analysis. Then we determine the backward error for the Sylvester equation, investigate its relationship with the residual, and derive a condition number. All these results respect the structure of the Sylvester equation and are relevant to any solution method. We also consider the special case of the Lyapunov equation and mention how the results extend to generalizations of the Sylvester equation.
15.1. Solving the Sylvester Equation One way to solve the Sylvester equation is to apply Gaussian elimination with partial pivoting (GEPP) to the “big” system (15.2), but the structure of the coefficient matrix cannot be exploited and the cost is a prohibitive O( m 3 n 3 ) flops. A more efficient method, requiring O( m 3 + n3) flops, is obtained with the aid of Schur decompositions of A and B. Let A and B have the real Schur decompositions A = URU T , (15.4) B = VSVT, where U and V are orthogonal and R and S are quasi-triangular, that is, block triangular with 1 × 1 or 2 × 2 diagonal blocks, and with any 2 × 2 diagonal blocks having complex conjugate eigenvalues. (See Golub and Van Loan [470, 1989, §7.4.1] for more details of the real Schur decomposition.) With the decompositions (15.4), the Sylvester equation transforms to (15.5) or, equivalently, Pz = d, where P = I n R – ST I m , z = vec(Z) and d = v e t(D). If R and S are both triangular then P is block triangular with triangular diagonal blocks, so Pz = d can be solved by substitution. Expressed in the notation of (15.5), the solution process take the form of n substitutions: if S is upper triangular then
Suppose now that R and S are quasi-triangular, and for definiteness assume that they are both upper quasi-triangular. Partitioning Z = (Zij ) con-
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formally with R = (Rij ) and S = (Sij ) we have (15.6) These equations can be used to determine the blocks of Z working up the block columns from first to last. Since Rii and Sjj are both of order 1 or 2, each system (15.6) is a linear system of order 1, 2, or 4 for Zij ; in the latter two cases it is usually solved by GEPP (or even Gaussian elimination with complete pivoting—see Problem 15.4). This Schur decomposition method for solving the Sylvester equation is due to Bartels and Stewart [74, 1972]. What can be said about its numerical stability? In the case where R and S are both triangular, Theorem 8.5 shows that (15.7) where cm,n denotes a constant depending on the dimensions m and n (in fact, we can take c m,n mn ). Thus which implies the weaker inequality (15.8) If R or S is quasi-triangular then the error analysis depends on how the systems of dimension 2 or 4 are solved. If GEPP followed by fixed precision iterative refinement is used for each of these systems and if for each system is not too ill conditioned and the vector is not too badly scaled, then (15.7) and (15.8) remain valid (see §11.2). Otherwise, we have only a normwise bound
Because the transformation of a matrix to Schur form (by the QR algorithm) is a backward stable process15 it is true overall that (15.9) Thus the relative residual is guaranteed to be bounded by a modest multiple of the unit roundoff u. Golub, Nash, and Van Loan [464, 1979] suggested a modification of the Bartels-Stewart algorithm in which A is reduced only to upper Hessenberg form: A = UHU T . The reduced system HZ – ZS = D can be solved by solving n systems that are either upper Hessenberg or differ from upper 15 See Golub and Van Loan [470, 19 8 9 , §7.5.6]. A proof is outside the scope of this book, but the necessary tools are Lemmas 18.3 and 18.8 about Householder and Givens rotations.
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Hessenberg form by the addition of an extra subdiagonal. As shown in [464, 1979 ], the Hessenberg–Schur algorithm has a smaller flop count than the Bartels-Stewart algorithm, with the improvement depending on the relative sizes of m and n. The computed solution again satisfies (15.9). The use of iterative methods to solve the Sylvester equation has attracted attention recently for applications where A and B are large and sparse [572, 1995 ], [588, 1992 ], [935, 1991 ], [1059, 19 88]. The iterations are usually terminated when an inequality of the form (15.9) holds, so here the size of the relative residual is known a priori (assuming the method converges). 15.2. Backward Error We saw in the last section that standard methods for solving the Sylvester equation are guaranteed to produce a small relative residual. Does a small relative residual imply a small backward error? The answer to this question for a general linear system is yes (Theorem 7. 1). But for the highly structured Sylvester equation the answer must be no, because for the special case of matrix inversion we know that a small residual does not imply a small backward error (§13.1). In this section we investigate the relationship between residual and backward error for the Sylvester equation. The normwise backward error of an approximate solution Y to (15.1) is defined by
(15.10) The tolerances a, β, and γ provide some freedom in how we measure the perturbations. Of most interest is the choice a = ||A||F, β = ||B||F, γ = ||C||F , for which we will call q the normwise relative backward error. The equation (A + ∆A)Y – Y(B + ∆B) = C + ∆C may be written (15.11) where the residual R = C – (AY – YB). A small backward error implies a small relative residual since, using the optimal perturbations from (15. 10) in (15.11), we have (15.12) To explore the converse question of what the residual implies about the backward error we begin by transforming (15.11) using the SVD of Y, Y = UΣ VT, where and are orthogonal and Σ = diag(σi ) 1 The numbers σ1 > σ 2 > . . . > σ m i n (m , n) > 0 are the singular values
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of Y and we define, in addition, σ m i n (m , n)+l = . . . = σ m a x (m , n) = 0. Equation (15.1 1) transforms to (15.13) where
This is an underdetermined system, with mn equations in m 2 + n 2 + mn unknowns. We can write it in the uncoupled form16 (15.14) For each i and j it is straightforward to show that the minimum of subject to (15.14) is attained for
These matrices minimize
Since η(Y ) is the minimum value of max{||a - 1 ∆ A||F,||β - 1∆ B||F,||γ - 1 ∆C||F} , it follows that . (15.15) where
(15.16) 16 For notational convenience we extend (if m < n) or (if m > n) to dimension m × n; the “fictitious” elements will be set to zero by the minimization.
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This expression shows that the backward error is approximately equal not to the normwise relative residual but to a componentwise residual corresponding to the diagonalized equation (15.13). From (15.15) and (15.16) we deduce that (15.17) where (15.18) The scalar µ > 1 is an amplification factor that measures by how much, at worst, the backward error can exceed the normwise relative residual. We now examine µ more closely, concentrating on the normwise relative backward error, for which a = ||A||F, β = ||B||F, and γ = ||C||F. First, note that if n = 1 and B = 0, so that the Sylvester equation reduces to a linear system Ay = c, then σ 1 = ||y||2 and σ k = 0 for k > 1, so and so we recover Theorem 7.1 (for the 2-norm) from (15.12) and (15.17), to within a constant factor. If m = n then (15.19) We see that µ is large only when (15.20) that is, when Y is ill conditioned and Y is a large-normed solution to the Sylvester equation. In the general case, with m n, one of and is always zero and hence µ can be large for a third reason: A (if m < n ) or B (if m > n) greatly exceeds the rest of the data in norm; in these cases the Sylvester equation is badly scaled. However, if we set a = β = ||A||F + ||B||F, which corresponds to regarding A and B as comprising a single set of data, then bad scaling does not affect µ. If we allow only A and B to be perturbed in (15.10) (as may be desirable if the right-hand side C is known exactly), then γ = 0 and (15.19) and (15.20) remain valid with ||C||F replaced by zero. In this case µ > ||Y||F||Y+|| 2 κ 2 (Y ) (for any m and n), so µ is large whenever Y is ill conditioned (and included in this case is matrix inversion). Conditions involving controllability which guarantee that the solution to the Sylvester equation with m = n is nonsingular are given by Hearon [508, 1977], while Datta [265, 1988] gives a determinantal condition for nonsingularity. It is an open problem to derive
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conditions for the Sylvester equation to have a well-conditioned solution (see Problem 15.5). The following numerical example illustrates the above analysis. This particular example was carefully chosen so that the entries of A and B are of a simple form, but equally effective examples are easily generated using random, ill-conditioned A and B of dimension m, n > 2. Let
Define C by the property that vet(C ) is the singular vector corresponding to the smallest singular value of In A – BT Im. With a = 10-6, we solved the Sylvester equation in MATLAB by the Bartels-Stewart algorithm and found that the computed solution X satisfies
Although has a very acceptable residual (as it must in view of (15.9)), its backward error is eight orders of magnitude larger than is necessary to achieve backward stability. We solved the same Sylvester equation using GEPP on the system (15.2). The relative residual was again less than u, but the backward error was appreciably larger: One conclusion we can draw from the analysis is that standard methods for solving the Sylvester equation are at best conditionally backward stable, since there exist rounding errors such that is the only nonzero element of and then (15.17) is an approximate equality, with µ possibly large.
15.2.1. The Lyapunov Equation If we put B = –AT in the Sylvester equation we obtain AX+ XA T = C, which is called the Lyapunov equation. This equation plays a major role in control and systems theory and it can be solved using the same techniques as for the Sylvester equation. If C = C T then C = AX + XA T = X T A T + AX T = C T , so X and T X are both solutions to the Lyapunov equation. If the Lyapunov equation is nonsingular (equivalently, for all i and j, by (15.3)) it therefore has a unique symmetric solution.
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We assume that C is symmetric and that Y is a symmetric approximate solution. The definition of backward error is now
The analogue of (15.11) is Let Y = UΛUT be a spectral decomposition, with A = diag residual equation transforms to
where written in uncoupled form as
and
Then the
This system can be
(15.21) We can obtain the minimum value of
by minimizing
subject to (15.21), for i, j = 1:n. The solution is
(Note that where
is symmetric since
is.) It follows that
where the last inequality is usually a good approximation. Comparing with (15.16) we see that respecting the extra structure of the Lyapunov equation has essentially no effect on the backward error. Finally, the analogue of (15.17) and (15.18) is
where
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15.3. Perturbation Result To derive a perturbation result we consider the perturbed Sylvester equation
which, on dropping second-order terms, becomes
This system may be written in the form (15.22) where P = In
A – BT
I m . If we measure the perturbations normwise by
where a, β, and γ are tolerances as in (15.10), then (15.23) is a sharp bound (to first order in
), where (15.24)
is the corresponding condition number for the Sylvester equation. The bound (15.23) can be weakened to (15.25) where
If then twice the upper bound in (15.25) can be shown to be a strict bound for the error. The perturbation bound (15.25) with a ||A||F, β = ||B||F, and γ = ||C||F is the one that is usually quoted in the literature for the Sylvester equation (see [464, 1979] and [522, 1988], for example), and corresponds to applying standard perturbation theory for Ax = b to (15.2). Note that ||P- 1||2 = sep(A, B)-1, where sep is the separation of A and B, (15.26)
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The sep function is an important tool for measuring invariant subspace sensitivity [470, 1989, §7.2.5], [940, 1973], [1050, 1979]. For the Lyapunov equation, a similar derivation to the one above shows that the condition number is (15.27) where Π is the vet-permutation matrix, which is defined by the property that vec(A T) = Π vet(A). How much can the bounds (15.23) and (15.25) differ? The answer is by an arbitrary factor. To show this we consider the case where B is normal (or equivalently, A is normal if we transpose the Sylvester equation). We can assume B is in Schur form, thus B = diag(µ j ) (with the µ j possibly complex). Then P = diag(A – ujjIm), and it is straightforward to show that if X = [x1, . . . , xn ], and if we approximate the 2-norms in the definitions of and Φ by Frobenius norms, then
while
These formulae show that in general Ψ and Φ will be of similar magnitude, and we know that Ψ < Φ from the definitions. However, Ψ can be much smaller than Φ. For example, suppose that γ = 0 and
Then if
we have Ψ 0, so and let X = (A – µ n n I m )Y, that A – µ nnIm = diag where Y = [y, y, . . . , y, 0] with ||(A–µ n n I m )y||2 = ||A–µ n n I m || 2 and ||y||2 = 1. Then, if γ =
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To summarize, the “traditional” perturbation bound (15.25) for the Sylvester equation can severely overestimate the effect of a perturbation on the data when only A and B are perturbed, because it does not take account of the special structure of the problem. In contrast, the perturbation bound (15.23) does respect the Kronecker structure, and consequently is attainable for any given A, B, and C. To obtain an a posteriori error bound for a computed solution ∆ X we can set ∆A = 0, ∆B = 0, and ∆C = AX – XB – C = R in (15.22), which leads to (15.28) A similar but potentially much smaller bound is described in the next section.
15.4. Practical Error Bounds For the Sylvester equation we can obtain an analogue of the practical error bound (7.27) by identifying Ax = b with (15.2). For the computed residual of a computed solution X we have
Therefore the bound is (15.29) where ||X|| := maxi,j |xij|. After transformation by the technique illustrated in (14.1), this bound can be estimated by the LAPACK norm estimator (Algorithm 14.4) at the cost of solving a few linear systems with coefficient matrices In A – B T Im and its transpose—in other words, solving a few Sylvester equations AX – XB = C and A T X – XB T = D. If the Bartels-Stewart algorithm is used, these solutions can be computed with the aid of the previously computed Schur decompositions of A and B. The condition number Ψ in (15.24) and sep(A, B) = can both be estimated in much the same way; alternatively, the power method can be used (see Ghavimi and Laub [440, 1995]). Other algorithms for efficiently estimating sep(A, B) given Schur decompositions of A and B are given by Byers [172, 1984] and Kågström and Poromaa [621, 1992]. The attraction of (15.29) is that large elements in the j th column of P- 1 may be countered by a small jth element of vec + vec(R u ), making the bound much smaller than (15.28). In this sense (15.29) has better scaling
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properties than (15.28), although (15.29) is not actually invariant under diagonal scalings of the Sylvester equation. We give a numerical example to illustrate the advantage of (15.29) over (15.28). Let where denotes a Jordan block of size n with eigenvalue Solving the Sylvester equation by the Bartels–Stewart algorithm we found that the bounds are
(where in evaluating (15.28) we replaced R by + Ru, as in (15.29)). Here, sep(A, B) = 1.67 × 10–16, and the bound (15.29) is small because relatively large columns of P-1 are nullified by relatively small elements of |vec + vec(R u ). For this example, with a = ||A||F, β = ||B||F, γ = ||C||F, we have Ψ = 7.00 × 109,
Φ = 1.70 × 1016,
confirming that the usual perturbation bound (15.25) for the Sylvester equation can be very pessimistic. Furthermore,
so we have an example where the backward error is small despite a large µ .
15.5. Extensions The Sylvester equation can be generalized in two main ways. One retains the linearity but adds extra coefficient matrices, yielding the generalized Sylvester equations (15.30) AXB + CXD = E and AX – YB = C, DX – YE = F.
(15.31)
These two forms are equivalent, under conditions on the coefficient matrices [210, 1987]; for example, defining Z := XB and W := –CX, (15.30) becomes AZ – WD = E, CZ + WB = 0. Applications of generalized Sylvester equations include the computation of stable eigendecompositions of matrix pencils [294, 198 7], [295, 1988], [622, 1993], [623, 1994] and the implementation of
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numerical methods for solving implicit ordinary differential equations [353, 19 8 0 ]. The second generalization incorporates a quadratic term, yielding the algebraic Riccati equation (15.32) This general Riccati equation and its symmetric counterpart with B = AT and F and G symmetric are widely used in control theory. The backward error results and perturbation theory of this chapter can be generalized in a straightforward way to (15.31) and (15.32). See Kågström [620, 1994] for (15.31) and Ghavimi and Laub [440, 1995] for (15.32). The backward error derivations do not extend to (15.30), because in this equation the coefficient matrices appear nonlinearly. A variation of the Lyapunov equation called the discrete-time Lyapunov equation has the form As in (15.30), the data appears nonlinearly. Ghavimi where and Laub [441, 1995] show how to derive an approximation to the backward error by linearizing an equation characterizing the optimal perturbations. Another generalization of the Sylvester equation, mainly of theoretical interest, is
where associated theory.
and
See Lancaster [685,
1970]
for
15.6. Notes and References This chapter is based on Higham [556, 1993]. The backward error derivations make use of ideas of Ghavimi and Laub [440, 1995]. The Sylvester equation is so named because Sylvester considered the homogeneous version of the equation [985, 1884]. Bhatia and Rosenthal [96, 1996] give a survey of theoretical results for the Sylvester equation in both finite- and infinite-dimensional spaces. For details of the role of the Sylvester equation in the eigenproblem see Bai, Demmel, and McKenney [38, 1993], [40, 1993] and the references therein. Iterative methods that make use of matrix inversion to solve the Sylvester equation are described by Miller [755, 1988] and Roberts [875, 1980]. Hammarling [496, 198 2] gives a method for solving the Lyapunov equation AX + XAT = –C in the case where A has eigenvalues with negative
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R EFERENCES
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real parts and C is positive semidefinite; his method directly computes the Cholesky factor of the solution (which is indeed symmetric positive definite— see Problem 15.2). A survey of the vec operator, the Kronecker product, and the vec-permutation matrix is given together with historical comments by Henderson and Searle [514, 1981]. Historical research by Henderson, Pukelsheim, and Searle [513, 1983] indicates that the Kronecker product should be called the Zehfuss product, in recognition of an 1858 paper by Zehfuss that gives a determinantal result involving the product. The vet-permutation matrix Π (which appears in (15.27)) is given explicitly by
and has the property that Applications of the Lyapunov equation in control theory, including special situations where an approximate solution of low rank is required, are discussed by Hodel [574, 1992]. A much older reference to applications is Barnett and Storey [68, 1968]. Algorithms and software for solving (15.30) are developed by Gardiner, Wette, Laub, Amato, and Moler [417, 1982], [418, 1982]. Perturbation theory for Lyapunov and Riccati equations can be found in the work of Byers [173, 1985], Hewer and Kenney [522, 1988], [650, 1990], and Gahinet, Laub, Kenney, and Hewer [411, 1990]. Chu [210, 1987] determines conditions for the existence of unique solutions to the generalized Sylvester equations (15.30) and (15.31). The appropriate conditions for (15.30) are that the pencils A + and D + are regular and the spectra of the pencils have an empty intersection, which neatly generalizes the conditions for the Sylvester equation to have a unique solution; the conditions for (15.31) are analogous. There is much work on algorithms and software for solving the algebraic Riccati equation. For a sampling, see Laub [691, 1979], Arnold and Laub [30, 1984], Byers [174, 1987], Gardiner, and Laub [416, 1991], and Kenney, Laub, and Papadopoulos [653, 1992]. An algorithm for estimating a generalization of sep that occurs in perturbation theory for the generalized Sylvester equation (15.31) is developed by Kågström and Westin [624, 1989]. Another generalization of the Sylvester equation is to take just one equation from (15.31), AX – YB = C ((15.13) is of this form). This equation can be underdetermined or overdetermined, depending on the dimensions of the coefficient matrices. Conditions involving generalized inverses that are both necessary and sufficient for the existence of a solution are given by Baksalary and Kala [49, 1979]. Zietak examines the inconsistent case [1130, 1985] for one
T HE S YLVESTER E QUATION
324
choice of dimensions giving an overdetermined system. Stewart [949, 1992] shows how to compute a minimum Frobenius norm least squares solution. The even more general equation AXB + CYD = E has also been analysed by Baksalary and Kala [50, 198 0], who again give necessary and sufficient conditions for the existence of a solution.
15.6.1. LAPACK The computations discussed in this chapter can all be done using LAPACK. The Bartels-Stewart algorithm can be implemented by calling xGEES to compute the Schur decomposition, using the level-3 BLAS routine xGEMM to transform the right-hand side C, calling xTRSYL to solve the (quasi-) triangular Sylvester equation, and using xGEMM to transform back to the solution X. The error bound (15.29) can be estimated using xLACON in conjunction with the above routines. A Fortran 77 code dggsvx [556, 1993] of Higham follows this outline and may appear in a future release of LAPACK. Routine xLASY2 solves a real Sylvester equation AX ± XB = σC in which A and B have dimension 1 or 2 and σ is a scale factor. It is called by xTRSYL . Kågström and Poromaa have developed codes for solving (15.31), which are intended for a future release of LAPACK [622, 1993], [623, 1994].
Problems 15.1. Show that the Sylvester equation AX – XA = I has no solution. 15.2. (Bellman [89,
1970,
§10.18]) Show that if the expression
exists for all C it represents the unique solution of the Sylvester equation AX + XB = C. (Hint: consider the matrix differential equation dZ/dt = AZ(t) + Z(t)B, Z(0) = C.) Deduce that the Lyapunov equation AX + XAT = –C has a symmetric positive definite solution if A has eigenvalues with negative real parts and C is symmetric positive definite. 15.3. (Byers and Nash [176,
1987])
Let
and consider
Show that there exists a minimizer X that is either symmetric or skewsymmetric. 15.4. How would you solve a Sylvester equation AX – XB = C in which A and B are of dimension 1 or 2? Compare your method with the one used in the LAPACK routine XLASY2 . 15.5. (RESEARCH PROBLEM) Derive conditions for the Sylvester equation to have a well-conditioned solution.
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Chapter 16
Stationary Iterative Methods
I recommend this method to you for imitation. You will hardly ever again eliminate directly, at least not when you have more than 2 unknowns. The indirect [iterative] procedure can be done while half asleep, or while thinking about other things.17 — CARL FRIEDRICH GAUSS, Letter to C. L. Gerling (1823)
The iterative method is commonly called the “Seidel process, ” or the “Gauss–Seidel process. ” But, as Ostrowski (1952) points out, Seidel (1874) mentions the process but advocates not using it. Gauss nowhere mentions it. — GEORGE E. FORSYTHE, Solving Linear Algebraic Equations Can Be Interesting (1953)
The spurious contributions in null(A) grow at worst linearly and if the rounding errors are small the scheme can be quite effective. — HERBERT B. KELLER, On the Solution of Singular and Semidefinite Linear Systems by Iteration (1965)
17 Gauss refers here to his relaxation method for solving the normal equations. The translation is taken from Forsythe [387, 1951 ].
325
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S TATIONARY I TERATIVE M ETHODS
Table 16.1. Dates of publication of selected iterative methods. Based on Young [1123, 19 8 9 ]. 1845 1874 1910 1938–1939 1940s 1952
Jacobi method Jacobi Gauss-Seidel method Seidel Richardson Richardson’s method Method of steepest descent Temple Various (analysis by Successive overrelaxation Young and Frankel) (SOR) method Hestenesand Stiefel Conjugate gradient method
Iterative methods for solving linear systems have along history, going back at least to Gauss. Table 16.1 shows the dates of publication of selected methods. It is perhaps surprising, then, that rounding error analysis for iterative methods is not well developed. There are two main reasons for the paucity of error analysis. One is that in many applications accuracy requirements are modest and are satisfied without difficulty, resulting in little demand for error analysis. Certainly there is no point in computing an answer to greater accuracy than that determined by the data, and in scientific and engineering applications the data often has only a few correct digits. The second reason is that rounding error analysis for iterative methods is inherently more difficult than for direct methods, and the bounds that are obtained are harder to interpret. In this chapter we consider a simple but important class of iterative methods, stationary iterative methods, for which a reasonably comprehensive error analysis can be given. The basic question that our analysis attempts to answer is “What is the limiting accuracy of a method in floating point arithmetic?” Specifically, how small can we guarantee that the backward or forward error will be over all iterations k = 1, 2 ,. . .? Without an answer to this question we cannot be sure that a convergence test of the form (say) will ever be satisfied, for any given value of As an indication of the potentially devastating effects of rounding errors we present an example constructed and discussed by Hammarling and Wilkinson [497, 1976]. Here, A is the 100 × 100 lower bidiagonal matrix with aii 1.5 and ai,i– 1 1, and bi 2.5. The successive overrelaxation (SOR) method is applied in MATLAB with parameter w = 1.5, starting with the rounded version of the exact solution x, given by xi = 1 – (–2/3) i . The forward errors and the co-norm backward errors are plotted in Figure 16.1. The SOR method converges in exact arithmetic, since the iteration matrix has spectral radius 1/2, but in the presence of rounding errors it diverges. The iterate has a largest element of order 1013, for
16.1 S URVEY
OF
ERROR ANALYSIS
327
Figure 16.1. SOR iteration. k > 238, and for k > 100, The divergence is not a result of ill conditioning of A, since The reason for the initial rapid growth of the errors in this example is that the iteration matrix is far from normal; this allows the norms of its powers to become very large before they ultimately decay by a factor 1/2 with each successive power. The effect of rounding errors is to cause the forward error curve in Figure 16.1 to level off near k = 100, instead of decaying to zero as it would in exact arithmetic. More insight into the initial behaviour of the errors can be obtained using the notion of pseudo-eigenvalues; see §17.3.
16.1. Survey of Error Analysis Before analysing stationary iterative methods, we briefly survey the published error analysis for iterative methods. For symmetric positive definite systems, Golub [468, 196 2] derives both statistical and nonstatistical bounds for the forward error and residual of the Richardson method. Benschop and Ratz [92, 1971] give a statistical analysis of the effect of rounding errors on stationary iteration, under the assumption that the rounding errors are independent random variables with zero mean. Lynn [719, 196 4] presents a statistical analysis for the SOR method with a symmetric positive definite matrix. Hammarling and Wilkinson [497, 1976] give a normwise error analysis for
328
S TATIONARY I TERATIVE M ETHODS
the SOR method. With the aid of numerical examples, they emphasize that while it is the spectral radius of the iteration matrix M – 1 N that determines the asymptotic rate of convergence, it is the norms of the powers of this matrix that govern the behaviour of the iteration in the early stages. This point is also explained by Trefethen [1017, 1992], using the tool of pseudospectra. Dennis and Walker [302, 198 4] obtain bounds for for stationary iteration as a special case of error analysis of quasi-Newton methods for nonlinear systems. The bounds in [302, 198 4] do not readily yield information about normwise or componentwise forward stability. Bollen [133, 1984] analyses the class of “descent methods” for solving Ax = b, where A is required to be symmetric positive definite; these are obtained by iteratively using exact line searches to minimize the quadratic function F(x) = (A-1 b – x)TA(A-1 b – x). The choice of search direction pk = b – Axk =: rk yields the steepest descent method, while pk = ej (unit vector), where |rk|j = gives the Gauss-Southwell method. Bollen shows that both methods are normwise backward stable as long as a condition of the form cn κ(A)u < 1 holds. If the pk are cyclically chosen to be the unit vectors e 1, e2, . . ., en then the Gauss–Seidel method results, but unfortunately no results specific to this method are given in [133, 1984]. Wozniakowski [1112, 1977] shows that the Chebyshev semi-iterative method is normwise forward stable but not normwise backward stable, and in [1113, 1978] he gives a normwise error analysis of stationary iterative methods. Some of the assumptions in [1113, 1978] are difficult to justify, as explained by Higham and Knight [563, 1993]. In [1114, 198 0] Wozniakowski analyses a class of conjugate gradient algorithms (which does not include the usual conjugate gradient method). He obtains a forward error bound proportional to κ(A)3/2 and a residual bound proportional to K(A), from which neither backward nor forward normwise stability can be deduced. We note that as part of the analysis in [1114, 198 0] Wozniakowski obtains a residual bound for the steepest descent method that is proportional to K(A), and is therefore much weaker than the bound obtained by Bollen [133, 1984]. Zawilski [1125, 1991] shows that the cyclic Richardson method for symmetric positive definite systems is normwise forward stable provided the parameters are suitably ordered. He also derives a sharp bound for the residual that includes a factor κ(A), and which therefore shows that the method is not normwise backward stable. Arioli and Romani [29, 1992] give a statistical error analysis of stationary iterative methods. They investigate the relations between a statistically defined asymptotic stability factor, ill conditioning of M- 1 A, where A = M – N is the splitting, and the rate of convergence. Greenbaum [479, 1989] presents a detailed error analysis of the conjugate gradient method, but her concern is with the rate of convergence rather than
16.2 FORWARD ERROR ANALYSIS
329
the attainable accuracy. An excellent survey of work concerned with the effects of rounding error on the conjugate gradient method (and the Lanczos method) is given by Greenbaum and Strakos in the introduction of [480, 1992]; see also Greenbaum [481, 1994]. Notay [799, 1993] analyses how rounding errors influence the convergence rate of the conjugate gradient method for matrices with isolated eigenvalues at the ends of the spectrum. Van der Vorst [1058, 1990] examines the effect of rounding errors on preconditioned conjugate gradient methods with incomplete Cholesky preconditioners. The analysis given in the remainder of this chapter is from Higham and Knight [563, 1993], [564, 1993], wherein more details are given. Error analysis of Kaczmarz’s row-action method is given by Knight [663, 1993].
16.2. Forward Error Analysis A stationary iterative method has the form
where is nonsingular and M is nonsingular. We assume that the spectral radius p(M - 1 N) < 1, so that in exact arithmetic the iteration converges for any starting vector x0. We are not concerned with the size of constants in this analysis, so we denote by cn a constant of order n. The computed vectors satisfy an equality of the form
which we write as (16.1) where We will assume that M is triangular (as is the case for the Jacobi, Gauss– Seidel, SOR, and Richardson iterations), so that and f k accounts solely for the errors in forming Hence (16.2) Solving the recurrence (16.1) we obtain (16.3) where G = M– 1 N. Since the iteration is stationary at x, (16.4)
S TATIONARY I TERATIVE M ETHODS
330 and so the error em+1 :=
satisfies (16.5)
We have (16.6) where µ k is the bound for ξk defined in (16.2). The first term, |Gm+ 1e0 |, is the error of the iteration in exact arithmetic and is negligible for large m. The accuracy that can be guaranteed by the analysis is therefore determined by the last term in (16.6), and it is this term on which the rest of the analysis focuses. At this point we can proceed by using further componentwise inequalities or by using norms. First we consider the norm approach. By taking norms in (16.6) and defining (16.7) we obtain
(16.8) where the existence of the sum is assured by the result of Problem 16.1. If = q < 1 then (16.8) yields
Thus if q is not too close to 1 (q < 0.9, say), and γx and are not too large, a small forward error is guaranteed for sufficiently large m. Of more interest is the following componentwise development of (16.6). Defining (16.9) so that
for all k, we have from (16.2), (16.10)
16.2 FORWARD ERROR ANALYSIS
331
Hence (16.6) yields (16.11) where, again, the existence of the sum is assured by the result of Problem 16.1. Since A = M – N = M(I – M- 1 N) we have
The sum in (16.11) is clearly an upper bound for |A - 1 |. Defining c(A) > 1 by
(16.12) we have our final bound (16.13) An interesting feature of stationary iteration methods is that if the elements of M and N are multiples of the elements in the corresponding positions of A, then any scaling of the form diagonal) leaves the eigenvalues of M– 1 N unchanged; hence the asymptotic convergence rate is independent of row and column scaling. This scale independence applies to the Jacobi and SOR iterations, but not, for example, to the stationary Richardson iteration, for which M = I. One of the benefits of doing a componentwise analysis is that under the above assumptions on M and N the bound (16.13) largely shares the scale independence. In (16.13) the scalar c(A) is independent of the row and column scaling of A, and the term |A– 1 |(|M| + |N|)|x| scales in the same way as x. Furthermore, θ x can be expected to depend only mildly on the row and column scaling, because the bound in (16.2) for the rounding error terms has the correct scaling properties. What can be said about c(A)? In general, it can be arbitrarily large. Indeed, c(A) is infinite for the Jacobi and Gauss–Seidel iterations for any n > 3 if A is the symmetric positive definite matrix with a ij = min(i, j), because A-1 is tridiagonal and (M - 1 N )k M–1 is not. If M-1 and M- 1N both have nonnegative elements then c(A) = 1; as we will see in the next section, this condition holds in some important instances. Some further insight into c(A) can be obtained by examining the case where M– 1 N is diagonal with eigenvalues It is easy to show that c(A) = maxi so c(A) can be large only if p(M– 1 N) is close to 1. Although M– 1 N cannot be diagonal for the Jacobi or Gauss–Seidel
332
S TATIONARY I TERATIVE M ETHODS
methods, this formula can be taken as being indicative of the size of c(A) when M– 1 N is diagonalizable with a well-conditioned matrix of eigenvectors. We therefore have the heuristic inequality, for general A, (16.14) In practical problems where stationary iteration is used, we would expect c(A) to be of modest size (O(n), say) for two reasons. First, to achieve a reasonable convergence rate, p(M– 1 N) has to be safely less than 1, which implies that the heuristic lower bound (16.14) for c(A) is not too large. Second, even if A is sparse, A–1 will usually be full, and so there are unlikely to be zeros on the right-hand side of (16.12). (Such zeros are dangerous because they can make c(A) infinite.) Note that in (16.13) the only terms that depend on the history of the iteration are |Gm+ 1e0 | and θx . In using this bound we can redefine x 0. to be any iterate thereby possibly reducing θx . This is a circular argument if used to obtain a priori bounds, but it does suggest that the potentially large θ x term will generally be innocuous. Note that if xi = 0 for some i then θ x is infinite unless = 0 for all k. This difficulty with zero components of x can usually be overcome by redefining
for which the above bounds remain valid if θx is replaced by 2θ x . Finally, we note that (16.13) implies (16.15) If θxc(A) = O(1) and |M| + |N| < a|A|, with a = O(1), this bound is of the form cn cond(A, x)u as m and we have componentwise forward stability. Now we specialize the forward error bound (16.15) to the Jacobi, Gauss– Seidel, and SOR iterations. 16.2.1. Jacobi’s Method For the Jacobi iteration, M = D = diag(A) and N = diag(A) – A. Hence |M| + |N| = |M – N| = |A|, and so (16.15) yields (16.16) If A is an M-matrix then M-1 > 0 and M-1 > 0, so c(A) = 1. Hence in this case we have componentwise forward stability as m if θx is suitably bounded.
16.2 FORWARD ERROR ANALYSIS
333
Table 16.2. Jacobi method, a = l/2 – 8- j .
j j j j j
= = = = =
1 2 3 4 5
p(M-1N) 0.75 0.97 0.996 1.00 1.00
Iters. 90 352 1974 11226 55412
cond(A,x) 3.40 4.76 4.97 5.00 5.00
2.22e-16 1.78-15 1.42e-14 1.14e-13 9.10e-13
1.27e-16 9.02e-16 7.12e-15 5.69e-14 4.55e-13
Table 16.3. Jacobi method, a = -(1/2-8 - j).
j j j j j
= = = = =
1 2 3 4 5
p(M - 1 N) 0.75 0.97 0.996 1.00 1.00
Wozniakowski [1113, matrix
Iters. 39 273 1662 9051 38294
1978,
cond(A,x) 7.00 6.30e1 5.11e2 4.09e3 3.28e4
4.44e-16 4.88e-15 4.22e-14 3.41e-13 2.73e-12
5.55e-17 7.63e-17 8.24e-17 8.32e-17 8.33e-17
Ex. 4.1] cites the symmetric positive definite
as a matrix for which the Jacobi method can be unstable, in the sense that there exist rounding errors such that no iterate has a relative error bounded by Let us see what our analysis predicts for this example. Straightforward manipulation shows that if a = 1/2 – then so as (The heuristic lower bound (16.14) is approximately in this case.) Therefore (16.16) suggests that the Jacobi iteration can be unstable for this matrix. To confirm the instability we applied the Jacobi method to the problem with x = [1, 1, 1] T and a = 1/2 – 8 – j, j = 1:5. We took a random x 0. with ||x – x0 ||2 = 10-10, and the iteration was terminated when there was no decrease in the norm of the residual for 50 consecutive iterations. Table 16.2 reports the smallest value of over all iterations, for each j; the number of iterations is shown in the column “Iters.” The ratio takes the values 8.02, 7.98, 8.02,
334
S TATIONARY I TERATIVE M ETHODS
7.98 for j = 1:4, showing excellent agreement with the behaviour predicted by (16.16), since Moreover, in these tests and setting the bound (16.16) is at most a factor 13.3 larger than the observed error, for each j. If –1/2 < a < 0 then A is an M-matrix and c(A) = 1. The bound (16.16) shows that if we set a = –( 1/2 – 8 – j) and repeat the above experiment then the Jacobi method will perform in a componentwise forward stable manner (clearly, is to be expected). We carried out the modified experiment, obtaining the results shown in Table 16.3. All the values are less than cond(A,x) u, so the Jacobi iteration is indeed componentwise forward stable in this case. Note that since p ( M – 1 N) and ||M- 1 N||2 take the same values for a and –a, the usual rate of convergence measures cannot distinguish between these two examples. 16.2.2. Successive Overrelaxation The SOR method can be written in the form Mxk+1
=
Nxk+ b , where
and where A = D + L + U, with L and U strictly lower triangular and upper triangular, respectively. The matrix |M| + |N| agrees with |A| everywhere except, possibly, on the diagonal, and the best possible componentwise inequality between these two matrices is (16.17) Note that f(w) = 1 for 1 < w < 2, and we have
as
From (16.15)
If A is an Al-matrix and 0 < w < 1 then M -1 > 0 and M- 1 N > 0, so c(A) = 1. The Gauss–Seidel method corresponds to w = 1, and it is interesting to note that for this method the forward error bound has exactly the same form as that for the Jacobi method (though c(A) and θ x are, of course, different for the two methods).
16.3. Backward Error Analysis We now turn our attention to bounding the residual vector, rk = b – From (16.3) and (16.4) we find that
16.3 B ACKWARD E RROR A NALYSIS
335
It is easy to show that AG k = H k A, where H := NM-1 (recall that G = M – 1 N). Therefore (16.18) Taking norms and using (16.2) gives, similarly to (16.8), (16.19) where
The following bound shows that σ is small if close to 1:
with q not too
A potentially much smaller bound can be obtained under the assumption that H is diagonalizable. If H = XDX –1, with D = diag then
(16.20) Note that so we see the reappearance of the term in the heuristic bound (16.14). The bound (16.20) is of modest size if the eigenproblem for H is well conditioned is small) and p(H) is not too close to 1. Note that real eigenvalues of H near +1 do not affect the bound for σ, even though they may cause slow convergence. To summarize, (16.19) shows that, for large m, the normwise backward error for the is certainly no larger than
Note that for the Jacobi and Gauss-Seidel methods, and also for the SOR method if w > 1.
336
S TATIONARY I TERATIVE M ETHODS
A componentwise residual bound can also be obtained, but it does not lead to any identifiable classes of matrix or iteration for which the componentwise relative backward error is small. To conclude, we return to our numerical examples. For the SOR example at the start of the chapter, c(A) = O(1045) and σ = O(1030), so our error bounds for this problem are all extremely large. In this problem maxi |1 – where so (16.14) is very weak; (16.20) is not applicable since M– 1 N is defective. For the first numerical example in §16.2.1, Table 16.2 reports the minimum backward errors For this problem it is straightforward to show that The ratios of backward errors for successive value of j are 7.10, 7.89, 7.99, 8.00, so we see excellent agreement with the behaviour predicted by the bounds. Table 16.3 reports the normwise backward errors for the second numerical example in §16.2.1. The backward errors are all less than u, which again is close to what the bounds predict, since it can be shown that σ < 5 for –1/2 < a < 0. In both of the examples of §16.2.1 the componentwise backward error and in our practical experience this behaviour is typical or the Jacobi and SOR iterations.
16.4. Singular Systems Singular linear systems occur in a variety of applications, including the computation of the stationary distribution vector in a Markov chain [94, 1994], [647, 1983] and the solution of a Neumann boundary value problem by finite difference methods [834, 1976]. Because of the structure and the possibly large dimension of the coefficient matrices in these applications, iterative solution methods are frequently used. An important question is how the rather delicate convergence properties of the iterative methods are affected by rounding errors. In this section we extend the analysis of stationary iteration to singular systems. 16.4.1. Theoretical Background A useful tool in analysing the behaviour of stationary iteration for a singular system is the Drazin inverse. This can be defined, for as the unique matrix AD such that
where k = index(A). The index of A is the smallest nonnegative integer k such that rank(Ak) = rank(A k+1); it is characterized as the dimension of the largest Jordan block of A with eigenvalue zero. If index(A) = 1 then AD
337
16.4 S INGULAR S YSTEMS
is also known as the group inverse of A and is denoted by A#. The Drazin inverse is an “equation-solving inverse” precisely when index(A) < 1, for then AADA = A, and so if Ax = b is a consistent system then ADb is a solution. As we will see, however, the Drazin inverse of the coefficient matrix A itself plays no role in the analysis. The Drazin inverse can be represented explicitly as follows. If
where P and B are nonsingular and N has only zero eigenvalues, then
Further details of the Drazin inverse can be found in Campbell and Meyer’s excellent treatise [180, 1979, Chap. 7]. Let be a singular matrix and consider solving Ax = b by stationary iteration with a splitting A = M – N, where M is nonsingular. First, we examine the convergence of the iteration in exact arithmetic. Since any limit point x of the sequence {x k } must satisfy Mx = Nx + b, or Ax = b, we restrict our attention to consistent linear systems. (For a thorough analysis of stationary iteration for inconsistent systems see Dax [269, 1990]. ) As in the nonsingular case we have the relation (cf. (16.4)): (16.21) m where G = M – 1N. Since A is singular, G has an eigenvalue 1, so G does not tend to zero as that is, G is not convergent. If the iteration must exist. Following Meyer and is to converge for all x0 then Plemmons [752, 1977], we call a matrix B for which exists semiconvergent. We assume from this point on that G is semiconvergent. It is easy to see [94, 1994, Lem. 6.9] that G must have the form
(16.22) where P is nonsingular and
< 1. Hence
To rewrite this limit in terms of G, we note that (16.23)
STATIONARY ITERATIVE METHODS
338 and, since
is nonsingular, (16.24)
Hence (16.25) To evaluate the limit of the second term in (16.21) we note that, since the system is consistent, M– 1 b = M-1 Ax = (I – G)x, and so
We note in passing that the condition that G is semiconvergent is equivalent to I – G having index 1, in view of (16.23), but that this condition does not imply that A = M(I – G) has index 1. The conclusion is that if G is semiconvergent, stationary iteration converges to a solution of AX = b that depends on x 0 : (16.26) The first term in this limit is in null(I – G) and the second term is in range(I – G). To obtain the unique solution in range(I – G) we should take for x0. any vector in range(I – G) (x0 = 0, say). 16.4.2. Forward Error Analysis We wish to bound em+1 = x – where x is the limit in (16.26) corresponding to the given starting vector x0. The analysis proceeds as in the nonsingular case, up to the derivation of equation (16.5):
As before, the first term, Gm+ 1e 0, is negligible for large m, because it is the error after m + 1 stages of the exact iteration and this error tends to zero. To obtain a useful bound for the second term, we cannot simply take norms or absolute values, because grows unfoundedly with m (recall that G has an eigenvalue 1). Our approach is to split the vectors according to where and
16.4 S INGULAR SYSTEMS
339
null(I – G); this is a well-defined splitting because range(I – G) and null(I – G) are complementary subspaces (since index(I – G) = 1, or equivalently, G is semiconvergent). Using the properties of the splitting the error can be written as
We achieve the required splitting for ξi via the formulae
where Hence the error can be written as (16.27) Clearly, as m the final term in this expression can become unbounded, but since it grows only linearly in the number of iterations it is unlikely to have a significant effect in applications where stationary iteration converges quickly enough to be of practical use. Now we bound the term (16.28) Using inequality (16.2) and the definition of γx in (16.7) and θx in (16.9), we have
(16.29) The convergence of the two infinite sums is assured by the result of Problem 16.1, since by (16.22)-(16.24), G i E = Gi (I – G)D (I – G)
(16.30)
340
S TATIONARY I TERATIVE M ETHODS
where We conclude that we have the normwise error bound
(16.31) On setting E = I we recover the result (16.8) for the nonsingular case. If we assume that Γ is diagonal, so that P in (16.30) is a matrix of eigenvectors of G, then
This bound shows that a small forward error is guaranteed if O(1) and the second largest eigenvalue of G is not too close to 1. (It is this subdominant eigenvalue that determines the asymptotic rate of convergence of the iteration.) Turning to the componentwise case, we see from (16.24) and (16.30) that
Because of the form of the sum in (16.29), this prompts us to define the scalar c(A) > 1 by
in terms of which we have the componentwise error bound
(16.32) Again, as a special case we have the result for nonsingular A, (16.13). To what should we compare this bound? A perturbation result for Ax = b is given in [564, 1993] that projects the perturbations of A and b into range (I – G) and thus can be thought of as gauging the effect of perturbations to the “nonsingular part of the system”. For perturbations of order it gives an expression Hence we can deduce conditions on a stationary iterative method that ensure it is componentwise forward stable, in the sense of yielding a solution whose
16.5 S TOPPING
AN
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error is no larger than the uncertainty in x caused by rounding the data. The constants θx and c(A) should be bounded by d n, where dn denotes a slowly should hold, as it growing function of n; the inequality |M| + |N| < does for the Jacobi method and for the SOR method when where β is positive and not too close to zero; and the “exact error” Gm + 1e 0 must decay quickly enough to ensure that the term (m + 1) | (I – E) M-1| does not grow too large before the iteration is terminated. Numerical results given in [564, 1993] show that the analysis can correctly predict forward and backward stability, and that for certain problems linear growth of the component of the error in null(A) can indeed cause an otherwise convergent iteration to diverge, even when starting very close to a solution.
16.5. Stopping an Iterative Method What convergence test should be used to stop an iterative linear equation solver? In this section we explain how backward errors and condition numbers help answer this question. Note first that most iterative methods for solving Ax = b compute all or part of a matrix–vector product w = Aυ on each iteration, and in floating point arithmetic we have
where m is the maximum number of nonzeros per row of A. The method therefore cannot distinguish between A and A + ∆A where |∆ A| < γm |A|, and so there is no point in trying to achieve a componentwise relative backward error less than γm . Of course, instability of a method (or simply lack of convergence) may pose further restrictions on how small a backward error can be achieved. It is worth keeping in mind throughout this discussion that in practical applications accuracy and stability requirements are often quite modest because of large errors or uncertainties in the data, or because the iteration is an “inner step” of some “outer” iterative process. Indeed, one of the advantages of iterative methods is that the slacker the convergence tolerance the less computational effort is required, though the relation between tolerance and work depends very much on the method. Natural stopping criteria for an iterative method are that some measure of backward error or forward error does not exceed a tolerance, We will assume that the residual r = b – Ay is available for each iterate y , and that norms of y, r, and A can be computed or estimated. If r is not computed directly, but is recurred by the method, as, for example, in the conjugate gradient method, then the norm of the computed residual may differ from that of the true residual by several orders of magnitude; clearly, this affects the way that the stopping tests are interpreted.
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From Theorem 7.1 we have the following equivalences, for any subordinate matrix norm: (16.33a) (16.33b) (16.33c) These inequalities remain true with norms replaced by absolute values (Theo rem 7.3), but to evaluate (16.33b) and (16.33c) a matrix–vector product |A||y| must be computed, which is a nontrivial expense in an iterative method. Of these tests, (16.33c) is preferred in general, assuming it is acceptable to perturb both A and b. Note the importance of including both ||A|| and ||y|| in the test on ||r||; a test ||r|| < though scale independent, does not bound any relative backward error. Test (16.33a) is commonly used in existing codes, but may be very stringent, and possibly unsatisfiable. To see why, note that the residual of the rounded exact solution fl(x) = x + ∆ x , |∆x| < u|x|, satisfies, for any absolute norm,
and
If A is ill conditioned and x is a large-normed solution (that is, ||x|| ||A - 1 ||||b||), so that ||b|| is close to its lower bound, then (16.33a) is much harder to satisfy than (16.33c). If the forward error is to be bounded, then, for a nonsingular problem, tests can be derived involving the residual and A–1: the equality x – y = A – 1 r leads to normwise and componentwise forward error bounds, such as ||x – y||/||y|| < ||A- 1 ||||r||/||y||. Since these bounds involve A–1) they are nontrivial to compute. Some iterative methods automatically produce estimates of the extremal eigenvalues, and hence of For large, sparse symmetric positive definite matrices ||A- 1 ||2 can be cheaply estimated using the Lanczos method. Another possibility is to use condition estimation techniques (Chapter 14). The discussion in this section has inevitably been very general. Other considerations in practice include detecting nonconvergence of a method (due to rounding errors or otherwise), adapting the tests to accommodate a preconditioned (the residual r provided by the method may now be that for the preconditioned system), and using a norm that corresponds to a quantity being minimized by the method (a norm that may be nontrivial to compute).
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16.6. Notes and References The Gauss–Seidel method was chosen by Wilkinson [1080, 1948] as an “example of coding” for the ACE. Speaking of experience at that time at the National Physical Laboratory, he explained that “In general, direct methods have been used on those equations which did not yield readily to relaxation, and hence those solved by direct methods have nearly always been of an illconditioned type”. Stationary iterative methods are relatively easy to program, although there are many issues to consider when complicated data structures or parallel machines are used. A good source of straightforward C, Fortran, and MATLAB implementations of the Jacobi, Gauss–Seidel, and SOR methods, and other nonstationary iterative methods, is the book Templates for the Solution of Linear Systems [70, 1994]; the software is available from netlib. The book contains theoretical discussions of the methods, together with practical advice on topics such as data structures and the choice of a stopping criterion. The choice of stopping criteria for iterative methods is also discussed by Arioli, Duff, and Ruiz [28, 1992]. An up-to-date textbook on iterative methods is Axelsson [34, 1994].
Problems 16.1. Show that if and p(B) < 1, then the series are both convergent, where ||·|| is any norm. 16.2. (Descloux [304,
where
1963])
and
Consider the (nonlinear) iterative process
satisfies (16.34)
for some G(a) = a.
and where ||ek|| < a for all k. Note that a must satisfy
(a) Show that
(b) Show that the sequence {x k} is bounded and its points of accumulation x satisfy
(c) Explain the practical relevance of the result of (b).
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Chapter 17
Matrix Powers
Unfortunately, the roundoff errors in the mth power of a matrix, say Bm, are usually small relative to ||B||m rather than ||Bm||. — CLEVE B. MOLER and CHARLES F. VAN LOAN, Nineteen Dubious Ways to Compute the Exponential of a Matrix (1978)
It is the size of the hump that matters: the behavior of ||p(A∆ t)n|| = ||p(A∆ t)t/∆ t|| for small but nonzero t. The eigenvalues and the norm, by contrast, give sharp information only about the limits t or t 0. — DESMOND J. HIGHAM and LLOYD N. TREFETHEN, Stiffness of ODES (1993)
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Powers of matrices occur in many areas of numerical analysis. One approach to proving convergence of multistep methods for solving differential equations is to show that a certain parameter-dependent matrix is uniformly “power bounded” [493, 1991 , §V.7], [862, 1992 ]. Stationary iterative methods for solving linear equations converge precisely when the powers of the iteration matrix converge to zero. And the power method for computing the largest eigenvalue of a matrix computes the action of powers of the matrix on a vector. It is therefore important to understand the behaviour of matrix powers, in both exact and finite precision arithmetic. It is well known that the powers Ak of tend to zero as k if p (A) < 1, where p is the spectral radius. However, this simple statement does not tell the whole story. Figure 17.1 plots the 2-norms of the first 30 powers of a certain 3 × 3 matrix with p(A) = 0.75. The powers do eventually decay, but initially they grow rapidly. (In this and other similar plots, k on the x -axis is plotted against ||fl(Ak)||2 on the y-axis, and the norm values are joined by straight lines for plotting purposes.) Figure 17.2 plots the 2-norms of the first 250 powers of a 14 × 14 nilpotent matrix C 14 discussed by Trefethen and Trummer [1020, 198 7] (see §17.2 for details). The plot illustrates the statement of these authors that the matrix is not power bounded in floating point arithmetic, even though its 14th power should be zero. These examples suggest two important questions. • For a matrix with p(A) < 1, how does the sequence {||Ak||} behave? In particular, what is the size of the “hump” max k ||Ak||? • What conditions on A ensure that fl(Ak)
0 as k
We examine these questions in the next two sections.
17.1. Matrix Powers in Exact Arithmetic In exact arithmetic the limiting behaviour of the powers of is determined by A’s eigenvalues. As already noted, if p(A) < 1 then Ak as k if p(A) > 1, ||Ak|| as In the remaining case of p(A) = 1, ||A k || if A has a defective eigenvalue such that Ak does not converge if A has a nondefective eigenvalue such that (although the norms of the powers may converge); otherwise, the only eigenvalue of modulus 1 is the nondefective eigenvalue 1 and Ak converges to a nonzero matrix. These statements are easily proved using the Jordan canonical form (17.1a)
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Figure 17.1. A typical hump for a convergent, nonnormal matrix.
Figure 17.2. Diverging powers of a nilpotent matrix, C 1 4 .
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where X is nonsingular and
(17.1b)
as where n 1 + n2 + . . . + ns = n. We will call a matrix for which Ak k (or equivalently, p(A) < 1) a convergent matrix. The norm of a convergent matrix can be arbitrarily large, as is shown trivially by the scaled Jordan block (17.2) with and a >> 1. While the spectral radius determines the asymptotic rate of growth of matrix powers, the norm influences the initial behaviour of the powers. The interesting result that p(A) = for any norm (see Horn and Johnson [580, 1985, p. 299], for example) confirms the asymptotic role of the spectral radius. This formula for p(A) has actually been considered as a means for computing it; see Wilkinson [1089, 1965, pp. 615–617] and Friedland [408, 1991]. An important quantity is the “hump” max k ||Ak||, which can be arbitrarily large for a convergent matrix. Figure 17.1 shows the hump for the 3 × 3 upper triangular matrix with diagonal entries 3/4 and off-diagonal entries 2; this matrix has 2-norm 3.57. The shape of the plot is typical of that for a convergent matrix with norm bigger than 1. Note that if A is normal (so that in (17.1a) J is diagonal and X can be taken to be unitary) we have ||Ak|| 2 = || ||2 = = p(A)k , so the problem of bounding ||Ak|| is of interest only for nonnormal matrices. The hump phenomenon arises in various areas of numerical analysis. For example, it is discussed for matrix powers in the context of stiff differential equations by D. J. Higham and Trefethen [529, 1993], and by Moler and Van Loan [775, 1978] for the matrix exponential eAt with t More insight into the behaviour of matrix powers can be gained by considering the 2 × 2 matrix (17.2) with and a > 0. We have
and (17.3)
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Figure 17.3. Infinity norms of powers of 2 × 2 matrix J in (17.2), for λ = 0.99 and a = 0 (bottom line) and a = 10 -k, k = 0:3. Hence
It follows that the norms of the powers can increase for arbitrarily many steps until they ultimately decrease. Moreover, because k–1 tends to zero quite slowly as k the rate of convergence of to zero can be much slower than the convergence of to zero (see (17.3)) when is close to 1. In other words, nontrivial Jordan blocks retard the convergence to zero. For this 2 × 2 matrix, the hump maxk is easily shown to be approximately
where this value being attained for Figure 17.3 displays the norms of the first 400 powers of the matrices with and a = 0,0.001,0.01,0.1,1. The size and location of the hump are complicated expressions even in this simple case. When we generalize to direct sums of larger Jordan blocks and incorporate a similarity transformation, giving (17.1a), the qualitative behaviour of the powers becomes too difficult to describe precisely. In the rest of this section we briefly survey bounds for ||Ak||. First, however, we comment on the condition number that appears
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in various bounds in this chapter. The matrix X in the Jordan form (17.1a) is by no means unique [413, 1959, pp. 220–221], [467, 1976]: if A has distinct eigenvalues (hence J is diagonal) then X can be replaced by XD , for any nonsingular diagonal D, while if A has repeated eigenvalues then X can be replaced by XT, where T is a block matrix with block structure conformal with that of J and which contains some arbitrary upper trapezoidal Toeplitz blocks. We adopt the convention that κ(X) denotes the minimum possible value of κ(X) over all possible choices of X. This minimum value is not known for general A, and the best we can hope is to obtain a good estimate of it. However, if A has distinct eigenvalues then the results in Theorems 7.5 and 7.7 on diagonal scalings are applicable and enable us to determine (an approximation to) the minimal condition number. Explicit expressions can be given for the minimal 2-norm condition number for n = 2; see Young [1122, 1971, §3.8]. A trivial bound is ||Ak|| < ||A||k. A sharper bound can be derived in terms of the numerical radius
which is the point of largest modulus in the field of values of A. It is not hard to show that ||A||2/2 < r(A) < ||A||2 [580, 1985, p. 331]. The (nontrivial) inequality r(Ak) < r(A)k [580, 1985, p. 333] leads to the bound
If A is diagonalizable then, from (17.1a), we have the bound (17.4) for any p-norm. (Since p(A) < ||A|| for any norm, we also have the lower bound p (A)k < ||Ak||p .) This bound is unsatisfactory for two reasons. First, by choosing A to have well-conditioned large eigenvalues and ill-conditioned small eigenvalues we can make the bound arbitrarily pessimistic (see Problem 17.1). Second, it models norms of powers of convergent matrices as monotonically decreasing sequences, which is qualitatively incorrect if there is a large hump. The Jordan canonical form can also be used to bound the norms of the powers of a defective matrix. If XJX-1 is the Jordan canonical form of δ- 1 A then (17.5) for all δ > 0. This is a special case of a result of Ostrowski [812, 1973, Thin. 20.1] and the proof is straightforward: We can write δ - 1 A = X(δ-1 D + M)X –1, where D = diag and M is the off-diagonal part of the Jordan
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form. Then A = X(D + δM)X–1, and (17.5) follows by taking norms. An alternative way of writing this bound is
where A = XJX–1 and D = Note that this is not the same X as in (17.5): multiplying A by a scalar changes κ(X) when A is not diagonalizable. Both bounds suffer from the same problems as the bound (17.4) for diagonalizable matrices. Another bound in terms of the Jordan canonical form (17.1) of A is given by Gautschi [430, 1953]. For convergent matrices, it can be written in the form (17.6) where p = and c is a constant depending only on A (c is not defined explicitly in [430, 1953]). The factor kp-1 makes this bound somewhat more effective at predicting the shapes of the actual curve than (17.5), but again c can be unsuitably large. Since the norm estimation problem is trivial for normal matrices, it is natural to look for bounds that involve a measure of nonnormality. Consider the Schur decomposition Q*AQ = D+N, where N is strictly upper triangular, and let S represent the set of all such N. The nonnormality of A can be measured by Henrici’s departure from normality [516, 1962]
For the Frobenius norm, Henrici shows that ||N||F is independent of the particular Schur form and that
László [690, 1994] has recently shown that ∆F(A) is within a constant factor of the distance from A to the nearest normal matrix:
where v(A) = min{||E|| F : A + E is normal}. Henrici uses the departure from normality to derive the 2-norm bounds
(17.7)
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. Empirical evidence suggests that the first bound in (17.7) can be very pessimistic. However, for normal matrices both the bounds are equalities. Another bound involving nonnormality is given by Golub and Van Loan [470, 1989, Lem. 7.3.2]. They show that, in the above notation,
for any θ > 0. This bound is an analogue of (17.5) with the Schur form replacing the Jordan form. Again, there is equality when A is normal (if we set θ = 0). To compare bounds based on the Schur form with ones based on the Jordan form we need to compare ∆(A) with κ(X). If A is diagonalizable then [710, 1969, Thin. 4]
it can be shown by a 2 × 2 example that minX κ 2 (X) can exceed ∆F(A)/||A||F by an arbitrary factor [201, 1993, §4.2.7], [190, 1996, §9.1.1]. Another tool that can be used to bound the norms of powers is the pseudospectrum of a matrix, popularized by Trefethen [1017, 1992], [1018]. The of is defined, for a given to be the set
and it can also be represented, in terms of the resolvent (zI – A) -1, as
As Trefethen notes [1017, 1992], by using the Cauchy integral representation of Ak (which involves a contour integral of the resolvent) one can show that (17.8) where the (17.9) This bound is very similar in flavour to (17.5). The difficulty is transferred from estimating κ(X) to choosing and estimating Bai, Demmel, and Gu [39, 1994] consider A with p (A) < 1 and manipulate the Cauchy integral representation of Ak in a slightly different way from Trefethen to produce a bound in terms of the distance to the nearest unstable matrix,
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Their bound is
where e < a m := (1+ 1/m) m +1 < 4. Note that d(A) < 1 when p (A) < 1, as is easily seen from the Schur decomposition. The distance d(A) is not easy to compute. One approach is a bisection technique of Byers [175, 1988]. Finally, we mention that the Kreiss matrix theorem provides a good estimate of supk >0 ||Ak|| for a general albeit in terms of an expression that involves–the resolvent and is not easy to compute:
where φ(A) = sup{(|z| – 1)||(zI – A)–1||2 : |z| > 1} and e = exp(1). Details and references are given by Wegert and Trefethen [1071, 1994].
17.2. Bounds for Finite Precision Arithmetic The formulae A · Ak or Ak · A can be implemented in several ways, corresponding to different loop orderings in each individual product, but as long as each product is formed using the standard formula (AB)ij = all these variations satisfy the same rounding error bounds. We do not analyse here the use of the binary powering technique, where, for example, A9 is formed as A((A2)2)2; alternate multiplication on the left and right (fl(A k ) = fl(Afl(A k-2 )A)); or fast matrix multiplication techniques such as Strassen’s method. None of these methods is equivalent to repeated multiplication in finite precision arithmetic. We suppose, without loss of generality, that the columns of Am are computed one at a time, the jth as fl(A(A(. . . (Aej) . . .))), where ej is the jth unit vector. The error analysis for matrix–vector multiplication shows that the jth computed column of Am satisfies (17.10) where, for both real and complex matrices, we have (Problem 3.7) (17.11) It follows that and so a sufficient condition for convergence of the computed powers is that (17.12)
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This result is useful in certain special cases: p(|A|) = p(A) if A is triangular or has a checkerboard sign pattern (since then |A| = DAD –1 where D = diag(±1)); if A is normal then p(|A| < (this bound being attained for a Hadamard matrix); and in Markov processes, where the aij are transition probabilities, |A| = A. However, in general p (|A|) can exceed p(A) by an arbitrary factor (see Problem 17.2). To obtain sharper and more informative results it is necessary to use more information about the matrix. In the following theorem we give a sufficient condition, baaed on the Jordan canonical form, for the computed powers of a matrix to converge to zero. Although the Jordan form is usually eschewed by numerical analysts because of its sensitivity to perturbations, it is convenient to work with in this application and leads to an informative result. Theorem 17.1 (Higham and Knight). Let with the Jordan form (17.1) have spectral radius p(A) < 1. A sufficient condition for fl(Am) as m is (17.13) where t = maxi ni . Proof. It is easy to see that if we can find a nonsingular matrix S such that (17.14) for all i, then the product
tends to 0 as m In the rest of the proof we construct such a matrix S for the ∆Ai in (17.10).
Now consider the matrix Its ith diagonal block is of the form where the only nonzeros in N are 1s on the first superdiagonal, and so
Defining S =
we have
and (17.15)
Now we set where 0 < θ < 1 and we determine θ so that (17.14) is satisfied, that is, so that for all i. From (17.11) and (17.15) we have
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Therefore (17.14) is satisfied if
that is, if If the integer t is greater than 1 then the function f ( θ ) = (1 – θ)t-1 θ has a maximum on [0,1] at θ* = t–1 and f(θ * ) = (t – 1)–1(1 – t– 1 ) t satisfies (4(t – 1))-1 < f(θ*) < e-1. We conclude that for all integers 1 < t < n.
is sufficient to ensure that (17.14) holds. If A is normal then ||A||2 = p(A) < 1, t = 1, and κ2 (X) = 1, so (17.13) can be written in the form
where cn is a constant depending on n. This condition is also easily derived by taking 2-norms in (17.10) and (17.11). We can show the sharpness of the condition in Theorem 17.1 by using the Chebyshev spectral differentiation matrix Cn described by Trefethen and Trummer [1020, 1987]. The matrix Cn arises from degree n – 1 polynomial interpolation of n arbitrary data values at n Chebyshev points, including a boundary condition at 1. It is nilpotent and is similar to a single Jordan block of dimension n. We generate Cn in MATLAB using the routine chebspec from the Test Matrix Toolbox (see Appendix E). Figure 17.4 shows the norms of the powers of four variants of the Cn matrix. The powers of C 8 converge to zero, while the powers of 15C 8 diverge. Using a technique for estimating k2 (X) described in [565, 1995], we find that 1.08 × 10–9, which is safely less than 1, so that Theorem 17.1 predicts convergence. For 15C 8 we have 2.7, so the theorem correctly does not predict convergence. Next, for the matrix A = C13 + 0.36I, whose powers diverge, we have 13.05, and for A = C13 + 0.01I, whose powers converge, 0.01, so again the theorem is reasonably sharp. The plots reveal interesting scalloping patterns in the curves of the norms. For C8 and 15C8 the dips are every 8 powers, but the point of first dip and the dipping intervals are altered by adding different multiples of the identity matrix, as shown by the C13 examples. Explaining this behaviour is an open problem (see Problem 17.3).
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Figure 17.4. Computed powers of chebspec matrices. We saw in the last section that the powers of A can be bounded in terms of the pseudospectral radius. Can the pseudospectrum provide information about the behaviour of the computed powers? Figure 17.5 shows approximations to the for the matrices used in Figure 17.4, where the (computed) eigenvalues are plotted as crosses “×”. We see that the pseudospectrum lies inside the unit disc precisely when the powers converge to zero. A heuristic argument based on (17.10) and (17.11) suggests that, if for randomly chosen perturbations ∆Ai with ||∆Ai || < cnu||A||, most of the eigenvalues of the perturbed matrices lie outside the unit disc, then we can expect a high percentage of the terms A + ∆Ai in (17.10) to have spectral radius bigger than 1 and hence we can expect the product to diverge. On the other hand, if the cnu||A||-pseudospectrum is wholly contained within the unit disc, each A + ∆Ai will have spectral radius less than 1 and the product can be expected to converge. (Note, however, that if p(A) < 1 and p (B) < 1 it is not necessarily the case that p(AB) < 1.) To make this heuristic precise, we need an analogue of Theorem 17.1 phrased in terms of the pseudospectrum rather than the Jordan form. Theorem 17.2 (Higham and Knight). Suppose that izable with A = and has a unique eigenvalue modulus. Suppose that and
is diagonalof largest where
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Figure 17.5. Pseudospectra for chebspec matrices. X -1 = (yij). If < 1 for = cnu||A||2, where cn is a constant depending only on n, then, provided that a certain term can be ignored, fl(Am ) = 0. Proof. It can be shown (see [565, 1995]) that the conditions on ||X|| 1 and imply there is a perturbation à = A + ∆A of A with ||∆ A|| 2 = such that
Hence, if < 1 then term and rearranging gives
Ignoring the
Using Theorem 17.1 we have the required result for cn = 4 n 2 (n + 2), since t = 1. Suppose we compute the eigenvalues of A by a backward stable algorithm, that is, one that yields the exact eigenvalues of A+E, where ||E||2 < cnu||A|| 2 , with cn a modest constant. (An example of such an algorithm is the QR algorithm [470, 1989, §7.5]). Then the computed spectral radius satisfies < In view of Theorem 17.2 we can formulate a rule of thumb-one
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that bears a pleasing symmetry with the theoretical condition for convergence: The computed powers of A can be expected to converge to 0 if the spectral radius computed via a backward stable eigensolver is less than 1. This rule of thumb has also been discussed by Trefethen and Trummer [1020, 19 8 7 ] and Reichel and Trefethen [866, 1992 ]. In our experience the rule of thumb is fairly reliable when is not too close to 1. For the matrices used in our examples we have, using MATLAB’S eig function,
and we observed convergence of the computed powers for C 8 and C13 + 0.01I and divergence for the other matrices.
17.3. Application to Stationary Iteration As we saw in the previous chapter, the errors in stationary iteration satisfy e k = (M - 1N) k e0, so convergence of the iteration depends on the convergence of (M – 1 N) k to zero as k While the errors in stationary iteration are not precisely modelled by the errors in matrix powering, because matrix powers are not formed explicitly, the behaviour of the computed powers fl((M - 1N) k ) can be expected to give some insight into the behaviour of stationary iteration. For the successive overrelaxation (SOR) example at the start of Chap= 0.5(–1) i – j . ter 16, the matrix G = M – 1N is lower triangular with gij The computed powers of G in MATLAB reach a maximum norm of 1028 at k = 99 and then decay to zero; the eventual convergence is inevitable in view of the condition (17.12), which clearly is satisfied for this triangular G. An approximation to the u||G||2-pseudospectrum is plotted in Figure 17.6, and we see clearly that part of the pseudospectrum lies outside the unit disk. These facts are consistent with the nonconvergence of the SOR iteration (see Figure 16.1). That the pseudospectrum of G gives insight into the behaviour of stationary iteration has also been observed by Trefethen [1015, 1990], [1017, 1992], [1018] and Chatelin and Frayssé [203, 1992], but no rigorous results about the connect ion are available.
17.4. Notes and References This chapter is baaed closely on Higham and Knight [565,
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Figure 17.6. Pseudospectrum for SOR iteration matrix. The analysis for the powers of the matrix (17.2) is modelled on that of Stewart [953, 1994], who uses the matrix to construct a Markov chain whose second largest eigenvalue does not correctly predict the decay of the transient. For some results on the asymptotic behaviour of the powers of a nonnegative matrix, see Friedland and Schneider [409, 1980]. Another application of matrix powering is in the scaling and squaring method for computing the matrix exponential, which uses the identity eA = (eA/m )m together with a Taylor or Padé approximation to eA/m ; see Moler and Van Loan [775, 1978].
Problems 17.1. Let be diagonalizable: A = XΛX -1, Λ = diag struct a parametrized example to show that the bound can be arbitrarily weak.
Con-
17.2. Show that p(|A|)/p(A) can be arbitrarily large for 17.3. (RESEARCH PROBLEM) Explain the scalloping patterns in the curves of norms of powers of a matrix, as seen, for example, in Figure 17.4. (Consider exact arithmetic, as the phenomenon is not rounding error dependent. ) 17.4. (RESEARCH PROBLEM) obtain a sharp sufficient condition for fl(Ak) 0 in terms of the Schur decomposition of with p(A) < 1.
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Chapter 18
QR Factorization
Any orthogonal matrix can be written as the product of reflector matrices. Thus the class of reflections is rich enough for all occasions and yet each member is characterized by a single vector which serves to describe its mirror. — BERESFORD N. PARLETT, The Symmetric Eigenvalue Problem (1980)
A key observation for understanding the numerical properties of the modified Gram–Schmidt algorithm is that it can be interpreted as Householder QR factorization applied to the matrix A augmented with a square matrix of zero elements on top. These two algorithms are not only mathematically . . . but also numerically equivalent. This key observation, apparently by Charles Sheffield, was relayed to the author in 1968 by Gene Golub. — AKE BJÖRCK, Numerics of Gram-Schmidt Orthogonalization (1994)
The great stability of unitary transformations in numerical analysis springs from the fact that both the and the Frobenius norm are unitarily invariant. This means in practice that even when rounding errors are made, no substantial growth takes place in the norms of the successive transformed matrices. — J. H. WILKINSON, Error Analysis of Transformations Based on the Use of Matrices of the Form I – 2wwH (1965)
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The QR factorization is a versatile computational tool that finds use in linear equation, least squares and eigenvalue problems. It can be computed in several ways, including by the use of Householder transformations and Givens rotations, and by the Gram–Schmidt method. We explore the numerical prop erties of all three methods in this chapter. We also examine the use of iterative refinement on a linear system solved with a QR factorization and consider the inherent sensitivity of the QR factorization.
18.1. Householder Transformations A Householder matrix (also known as a Householder transformation, or Householder reflector) is a matrix of the form
It enjoys the properties of symmetry and orthogonality, and, consequently, is involuntary (P2 = I). The application of P to a vector yields
Figure 18.1 illustrates this formula and makes it clear why P is sometimes called a Householder reflector: it reflects x about the hyperplane Householder matrices are powerful tools for introducing zeros into vectors. Consider the question “given x and y can we find a Householder matrix P such that Px = y?” Since P is orthogonal we clearly require that ||x|| 2 = ||y||2 . Now
and this last equation has the form aυ = x – y for some a. But P is independent of the scaling of v, so we can set a = 1. With υ = x – y we have
and, since xTx = yTy,
Therefore as required. We conclude that, provided ||x||2 = ||y||2, we can find a Householder matrix P such that Px = y. (Strictly speaking, we have to exclude the case x = y, which would require υ = 0, making P undefined).
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Figure 18.1. Householder matrix P times vector x.
Normally we choose y to have a special pattern of zeros. The usual choice is y = σe1 where σ = ±||x||2, which yields the maximum number of zeros in y. Then
We choose sign(σ) = –sign(x1) to avoid cancellation in the expression for υ 1 .
18.2. QR Factorization A QR factorization of
with m > n is a factorization
where is orthogonal and is upper triangular. The matrix R is called upper trapezoidal, since the term triangular applies only to square matrices. Depending on the context, either the full factorization A = QR or the “economy size” version A = Q 1 R 1 can be called a QR factorization. A quick existence proof of the QR factorization is provided by the Cholesky factorization: if A has full rank and ATA = RTR is a Cholesky factorization, then A = AR–1 · R is a QR factorization. The QR factorization is unique if A has full rank and we require R to have positive diagonal elements (A = QD · DR is a QR factorization for any D = diag(±1)). The QR factorization can be computed by premultiplying the given matrix by a suitably chosen sequence of Householder matrices. The process is
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illustrated for a generic 4 × 3 matrix as follows:
The general process is adequately described by the k th stage of the reduction to triangular form. With A1 = A we have, at the start of the kth stage,
where Rk-1 is upper triangular. Choose a Householder matrix and embed into an m × m matrix
such that
(18.2) Then let Ak+1 = PkAk. Overall, we obtain R = PnPn-1 . . . P1 A =: QTA (Pn = I if m = n). To compute Ak+1 we need to form We can write
which shows that the matrix product can be formed as a matrix–vector product followed by an outer product. This approach is much more efficient than forming explicitly and doing a matrix multiplication. The overall cost of the Householder reduction to triangular form is 2n 2 (m– n/3) flops. The explicit formation of Q requires a further 4(m 2 n–mn2 + n 3/3) flops, but for many applications it suffices to leave Q in factored form.
18.3. Error Analysis of Householder Computations It is well known that computations with Householder matrices are very stable. Wilkinson showed that the computation of a Householder vector, and the application of a Householder matrix to a given matrix, are both normwise stable, in the sense that the computed Householder vector is very close to
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the exact one and the computed update is the exact update of a tiny normwise perturbation of the original matrix [1089, 1965, pp. 153–162, 236], [1090, 19 6 5 ]. Wilkinson also showed that the Householder QR factorization algorithm is normwise backward stable [1089, p. 236]. In this section we give a combined componentwise and normwise error analysis of Householder matrix computations. The componentwise bounds provide extra information over the normwise ones that is essential in certain applications (for example, the analysis of iterative refinement). Lemma 18.1. Let Consider the following construction of and such that Px = σe1, where P = I – βvvT is a Householder matrix with β = 2/(vTv):
In floating point arithmetic the computed and
and
satisfy
where |θk| < γk . Proof. We sketch the proof. Each occurrence of δ denotes a different number bounded by |δ| < u. We compute fl(xTx) = (1 + θ n ) x T x, and then (the latter term 1 + θn +1 is suboptimal, but our main aim is to keep the analysis simple). Hence For notational convenience, define w = υ 1 + s. We have (essentially because there is no cancellation in the sum). Hence
For convenience we will henceforth write Householder matrices in the form I – v v T, which requires ||υ||2 = and amounts to redefining υ :=
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and β := 1 in the representation of Lemma 18.1. We can then write, using Lemma 18.1, (18.3) where, as required for the next two results, the dimension is now m. Here, we have introduced the generic constant
in which c denotes a small integer constant whose exact value is unimportant. We will make frequent use of γcm in the rest of this chapter, because it is not worthwhile to evaluate the integer constants in our bounds explicitly. Because we are not “chasing constants” we can afford to be somewhat cavalier and freely use inequalities such as
(recall Lemma 3.3), where we signify by the prime that the constant c on the right-hand side is different from that on the left. The next result describes the application of a Householder matrix to a vector, and is the basis of all the subsequent analysis. In the applications of interest P is defined as in Lemma 18.1, but we will allow P to be an arbitrary Householder matrix. Thus v is an arbitrary, normalized vector, and the only assumption we make is that the computed satisfies (18.3). Lemma 18.2. Let when
and consider the computation of y = satisfies (18.3): The computed
where P = I - vvT. Proof. (Cf. the proof of Lemma 3.8,) We have
where
and |∆b| < γm|b|. Hence
where |∆w| < γc m |υ||υT||b|. Then
We have
= (I – satisfies
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But then
Next, we consider a sequence of Householder transformations applied to a matrix. Again, each Householder matrix is arbitrary and need have no connection to the matrix to which it is being applied. In the cases of interest, the Householder matrices Pk have the form (18.2), and so are of ever-decreasing effective dimension, but to exploit this property would not lead to any significant improvement in the bounds. For the remaining results in this section, we make the (reasonable) assumption that an inequality of the form (18.4) holds, where r is the number of Householder transformations and it is implicit that G and H denote nonnegative matrices. Lemma 18.3. Consider the sequence of transformations
where A1 = and Pk = is a Householder matrix. Assume that the transformations are performed using computed Householder vectors that satisfy (18.3). The computed matrix Ar+1 satisfies (18.5) where Q T = PrPr–1 . . . P1 and ∆A satisfies the normwise and componentwise bounds
(In fact, we can take G = m- 1 eeT, where e = [1, 1,..., 1]T .) In the special case n = 1, so that A a, we have with Proof. First, we consider the jth column of A, aj, which undergoes the transformations By Lemma 18.2 we have
where each ∆Pk depends on j and satisfies ||∆ Pk||F < γcm. Using Lemma 3.6 we obtain
(18.6)
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using Lemma 3.1 and assumption (18.4). Hence ∆A in (18.5) satisfies
NOW, since ||aj||2 < ||aj||1 = eT|aj|, from (18.6) we have
where ||G|| F = 1 (since ||ee T || F = m). Finally, if n = 1, so that A is a column vector, then (as in the proof of Lemma 18.2) we can rewrite (18.5) where as Note that the componentwise bound for AA in Lemma 18.3 does not imply the normwise one, because of the extra factor m in the componentwise bound. This is a nuisance, because it means we have to state both bounds in this and other analyses. We now apply Lemma 18.3 to the computation of the QR factorization of a matrix. Theorem 18.4. Let be the computed upper trapezoidal QR factor (m > n) obtained via the Householder QR algorithm. Then of there exists an orthogonal such that
where ||∆A||F < nγ cm||A||F and |∆ A| < mnγ cmG|A| , with ||G||F = 1. The matrix Q is given explicitly as Q = (PnPn–1 . . . P1 )T, where Pk is the Householder matrix that corresponds to the exact application of the kth step of the algorithm to Ak. Proof. This is virtually a direct application of Lemma 18.3, with Pk defined as the Householder matrix that produces zeros below the diagonal in the kth column of the computed matrix Ak. one subtlety is that we do not explicitly compute the lower triangular elements of R, but rather set them to zero explicitly. However, it is easy to see that the conclusions of Lemmas 18.2 and 18.3 are still valid in these circumstances; the essential reason is that the elements of ∆Pb in Lemma 18.2 that correspond to elements that are zeroed by the Householder matrix P are forced to be zero, and hence we can set the corresponding rows of ∆P to zero too, without compromising the bound on ||∆P||F. Finally, we consider use of the QR factorization to solve a linear system. Given a QR factorization of a nonsingular matrix a linear system Ax = b can be solved by forming Q T b and then solving Rx = Q T b.
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From Theorem 18.4, the computed is guaranteed to be nonsingular if We give only componentwise bounds. Theorem 18.5. Let be nonsingular. Suppose we solve the system Ax = b with the aid of a QR factorization computed by the Householder algorithm. The computed satisfies
where
Proof. By A Theorem 18.4, the computed upper triangular factor satisfies A + ∆A = QR with |∆ A| < n 2 γ c n G 1 |A| and ||G1 ||F = 1. BY Lemma 18.3, the computed transformed right-hand side satisfies with Importantly, the same orthogonal matrix Q appears in the equations involving and By Theorem 8.5, the computed solution to the triangular system satisfies Premultiplying by Q yields
that is, we have
where
Using
where G > G 1 and ||G||F = 1. The proof of Theorem 18.5 naturally leads to a result in which b is perturbed. However, we can easily modify the proof so that only A is perturbed: the trick is to use Lemma 18.3 to write where and to premultiply by (Q + ∆Q)-T instead of Q in the middle of the proof. This leads to the result (18.7) An interesting application of Theorem 18.5 is to iterative refinement, as explained in §18.6.
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18.4. Aggregated Householder Transformations In Chapter 12 we noted that LU factorization algorithms can be partitioned so as to express the bulk of the computation as matrix-matrix operations (level3 BLAS). For computations with Householder transformations the same goal can be achieved by aggregating the transformations. This technique is widely used in LAPACK. One form of aggregation is the “WY” representation of Bischof and Van Loan [105, 1987]. This involves representing the product Qr = PrPr -1 . . . P1 of r Householder transformations (where = 2). in the form This can be done using the recurrence (18.8) Using the WY representation, a partitioned QR factorization can be developed as follows. Partition as (18.9) and compute the Householder QR factorization of A 1 ,
The product PrPr–1 . . . P1 = I + is accumulated using (18.8), as the Pi are generated, and then B is updated according to
which involves only level-3 BLAS operations. The process is now repeated on the last m – r rows of B. When considering numerical stability, two aspects of the WY representation need investigating: its construction and its application. For the construction, we need to show that satisfies (18.10) (18.11) for modest constants d 1 , d 2, and d 3. Now
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But this last equation is essentially a standard multiplication by a Householder matrix, albeit with less opportunity for rounding errors. It follows from Lemma 18.3 that the near orthogonality of is inherited by the condition on in (18.11) follows similarly and that on is trivial. Note that the condition (18. 10) implies that (18.12) that is, is close to an exactly orthogonal matrix (see Problem 18.13). Next we consider the application of Suppose we form for the B in (18.9), so that
Analysing this level-3 BLAS-based computation using (18.12) and the very general assumption (12.3) on matrix multiplication (for the 2-norm), it is straightforward to show that
(18.13) This result shows that the computed update is an exact orthogonal update of a perturbation of B, where the norm of the perturbation is bounded in terms of the error constants for the level-3 BLAS. Two conclusions can be drawn. First, algorithms that employ the WY rep resentation with conventional level-3 BLAS are as stable as the corresponding point algorithms. Second, the use of fast BLAS3 for applying the updates affects stability only through the constants in the backward error bounds. The same conclusions apply to the more storage-efficient compact WY representation of Schreiber and Van Loan [905, 1989], and the variation of Puglisi [848, 1992].
18.5. Givens Rotations Another way to compute the QR factorization is with Givens rotations. A Givens rotation (or plane rotation) G(i, j, θ) is equal to the identity matrix except that
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Figure 18.2. Givens rotation, y = G(i, j, θ)x. where c = cosθ and s = sinθ. The multiplication y = G(i, j, θ)x rotates x through θ radians clockwise in the (i, j) plane; see Figure 18.2. Algebraically,
and so yj = 0 if (18.14) Givens rotations are therefore useful for introducing zeros into a vector one at a time. Note that there is no need to work out the angle θ, since c and s in (18.14) are all that are needed to apply the rotation. In practice, we would scale the computation to avoid overflow (cf. §25.8). To compute the QR factorization, Givens rotations are used to eliminate the elements below the diagonal in a systematic fashion. Various choices and orderings of rotations can be used; a natural one is illustrated as follows for a generic 4 x 3 matrix:
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The operation count for Givens QR factorization of a general m x n matrix (m > n) is 3n 2 (m – n/3) flops, which is 50% more than that for Householder QR factorization. The main use of Givens rotations is to operate unstructured matrices—for example, to compute the QR factorization of a tridiagonal or Hessenberg matrix, or to carry out delicate zeroing in updating or downdating problems [470, 1989, §12.6]. Error analysis for Givens rotations is similar to that for Householder matrices—but a little easier. We omit the (straightforward) proof of the first result. Lemma 18.6. Let a Givens rotation G(i, j, θ) be instructed according to (18.14). The computed and satisfy (18.15) where Lemma 18.7. Let and consider the computation of y = is a computed Givens rotation in the (i, j) plane for which and (18.15). The computed satisfies
where satisfy
where Gij is an exact Givens rotation based on c and s in (18.15). All the rows of ∆Gij except the ith and jth are zero. Proof. The vector
where
so that
differs from x only in elements i and j. We have
and similarly for
Hence
We take
For the next result we need the notion of disjoint Givens rotations. Rotations Gi1 ,j 1, . . . , Gir,jr are disjoint if is js and js jt for s t . Disjoint rotations are “nonconflicting” and therefore commute; it matters neither mathematically nor numerically in which order the rotations are applied. (Disjoint
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rotations can therefore be applied in parallel, though that is not our interest here. ) Our approach is to take a given sequence of rotations and reorder them into groups of disjoint rotations. The reordered algorithm is numerically equivalent to the original one, but allows a simpler error analysis. As an example of a rotation sequence already ordered into disjoint groups, consider the following sequence and ordering illustrated for a 6 × 5 matrix:
Here, an integer k in position (i, j) denotes that the (i, j) element is eliminated on the kth step by a rotation in the (j, i) plane, and all rotations on the kth step are disjoint. For an m × n matrix with m > n there are r = m + n — 2 stages, and the Givens QR factorization can be written as Wr Wr-1 . . . W1 A = R, where each Wi is a product of at most n disjoint rotations. It is easy to see that an analogous grouping into disjoint rotations can be done for the scheme illustrated at the start of this section. Lemma 18.8. Consider the sequence of transformations A k+1 = WkAk,
k = 1:r,
where A1 = and each Wk is a product of disjoint Givens rotations. Assume that the individual Givens rotations are performed using computed sine and cosine values related to the exact values defining the Wk by (18.15). Then the computed matrix Âr+1 satisfies Âr+1 = QT(A + ∆A), where QT = Wr Wr–1 . . . W1 and ∆A satisfies the normwise and componentwise bounds
(In fact, we can take G = m - 1 eeT, where e = [1, 1, . . . , 1] T.) In the special case n = 1, so that A = a, we have â(r + 1) = (Q + ∆Q)T a with ||∆ Q||F < γ c r . Proof. The proof is analogous to that of Lemma 18.3, so we offer only a sketch. First, we consider the jth column of A, aj, which undergoes the
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transformations = W r . . . W1 aj. By Lemma 18.7 and the disjointness of the rotations, we have
where each ∆Wk depends on j and satisfies ||∆ Wk|| 2 < we obtain
Using Lemma 3.6
(18.16) Hence
The inequalities (18. 16) for j = 1:n imply that
where ||G||F = 1. The result for n = 1 is proved as in Lemma 18.3. II We are now suitably equipped to give a result for Givens QR factorization. Theorem 18.9. Let be the computed upper trapezoidal QR factor (m > n) obtained via the Givens QR algorithm, with any of standard choice and ordering of rotations. Then there exists an orthogonal such that with ||∆A||F < γc(m+n)||A||F and |∆ A| < mγ c(m+n)G|A|, ||G||F = 1. (The matrix Q is a product of Givens rotations, the kth of which corresponds to the exact application of the kth step of the algorithm to Âk.) It is interesting that the error bounds for QR factorization with Givens rotations are a factor n smaller than those for Householder QR factorization. This appears to be an artefact of the analysis, and we are not aware of any difference in accuracy in practice.
18.6. Iterative Refinement Consider a nonsingular linear system Ax = b, where Suppose we solve the system using a QR factorization A = QR computed using Householder or Givens transformations (thus, x is obtained by solving Rx = QTb). Theorem 18.5, and its obvious analogue for Givens rotations, show that satisfies (18.17)
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where ||G||F = 1 and p is a low-degree polynomial. Hence has a normwise backward error of order p(n)u. However, since G is a full matrix, (18.17) suggests that the componentwise relative backward error need not be small. In fact, we know of no nontrivial class of matrices for which Householder or Givens QR factorization is guaranteed to yield a small componentwise relative backward error. Suppose that we carry out a step of fixed precision iterative refinement, to obtain The form of the bound (18.17) enables us to invoke Theorem 11.4. We conclude that the componentwise relative backward error after one step of iterative refinement will be small as long as A is not too ill conditioned and is not too badly scaled. This conclusion is similar to that for Gaussian elimination with partial pivoting (GEPP), except that for GEPP there is the added requirement that the LU factorization not suffer large element growth. Recall from (18.7) that we also have a backward error result in which only A is perturbed. The analysis of $11.1 is therefore applicable, and analogues of Theorems 11.1 and 11.2 hold in which η = The performance of QR factorization with fixed precision iterative refinement is illustrated in Tables 11. 1–1 1.3 in §11.2. The performance is as predicted by the analysis. Notice that the initial componentwise relative backward error is large in Table 11.2 but that iterative refinement successfully reduces it to the roundoff level (despite being huge). It is worth stressing that the QR factorization yielded a small normwise relative backward error in each example in fact), as we know it must.
18.7. Gram–Schmidt Orthogonalization The oldest method for computing a QR factorization is the Gram-Schmidt orthogonalization method. It can be derived directly from the equation A = (Gram-Schmidt does not compute QR, where A, and the m x m matrix Q in the full QR factorization and hence does not provide a basis for the orthogonal complement of range(A)). Denoting by a j and q j the jth columns of A and Q, respectively, we have
yields, since Q has orthonormal columns, Premultiplying by i = 1:j — 1. Further,
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where
Hence we can compute Q and R a column at a time. To ensure that rjj > 0 we require that A has full rank. Algorithm 18.10 (classical Gram–Schmidt). Given of rank n this algorithm computes the QR factorization A = QR, where Q is m × n and R is n × n, by the Gram-Schmidt method. for j = 1:n for i = 1:j –1 end
Cost: 2mn2 flops (2n 3/3 flops more than Householder QR factorization with Q left in factored form). In the classical Gram–Schmidt method (CGS), aj appears in the computation only at the jth stage. The method can be rearranged so that as soon as qj is computed, all the remaining vectors are orthogonalized against qj. This gives the modified Gram-Schmidt method (MGS). Algorithm 18.11 (modified Gram-Schmidt). Given of rank n this algorithm computes the QR factorization A = QR, where Q is m × n and R is n × n, by the MGS method.
end end Cost: 2mn2
flops.
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It is worth noting that there are two differences between the CGS and MGS methods. The first is the order in which the calculations are performed: in the modified method each remaining vector is updated once on each step instead of having all its updates done together on one step. This is purely a matter of the order in which the operations are performed. Second, and more crucially in finite precision computation, two different (but mathematically equivalent ) formulae for rkj are used: in the classical method, rkj = which involves the original vector a j, whereas in the modified method a j is replaced in this formula by the partially orthogonalized vector Another way to view the difference between the two Gram–Schmidt methods is via representations of an orthogonal projection; see Problem 18.7. The MGS procedure can be expressed in matrix terms by defining Ak = MGS transforms A1 = A into An+1 = Q by the sequence of transformations Ak = Ak+ 1Rk, where Rk is equal to the identity except in the kth row, where it agrees with the final R. For example, if n = 4 and k = 3,
Thus R = Rn . . . R 1 . The Gram-Schmidt methods produce Q explicitly, unlike the Householder and Givens methods, which hold Q in factored form. While this is a benefit, in that no extra work is required to form Q, it is also a weakness, because there is nothing in the methods to force the computed to be orthonormal in the face of roundoff. Orthonormality of Q is a consequence of orthogonality relations that are implicit in the methods, and these relations may be vitiated by rounding errors. Some insight is provided by the case n = 2, for which the CGS and MGS methods are identical. Given a1, a2 we compute q 1 = a 1 /||a 1 ||2, which we will suppose is done exactly, and then we form the unnormalized vector q2 = a2 – The computed vector satisfies
where
Hence
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and so the normalized inner product satisfies (18.18) where is the angle between a 1 and a2. But where A = [a 1, a2] (Problem 18.8). Hence, for n = 2, the loss of orthogonality can be bounded in terms of κ 2 (A). The same is true in general for the MGS method, as proved by Björck [107, 196 7]. A direct proof is quite long and complicated, but a recent approach of Björck and Paige [119, 1992] enables a much shorter derivation; we take this approach here. The observation that simplifies the error analysis of the MGS method is that the method is equivalent, both mathematically and numerically, to Householder QR factorization of the padded matrix To understand this equivalence, consider the Householder QR factorization (18.19) Let q1 , be the vectors obtained by applying the MGS method to A. Then it is easy to see that
and that the multiplication A 2 = carries out the first step of the MGS method on A, producing the first row of R and
The argument continues in the same way, and we find that (18.20) With the Householder–MGS connection established, we are ready to derive error bounds for the MGS method by making use of our existing error analysis for the Householder method. Theorem 18.12. Suppose the MGS method is applied to n, yielding computed matrices and constants ci = ci (m, n) such that
of rank Then there are (18.21) (18.22)
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and there exists an orthonormal matrix Q such that (18.23) Proof. To prove (18.21) we use the matrix form of the MGS method. For the computed matrices we have
Expanding this recurrence, we obtain
Hence (18.24) and a typical term has the form (18.25) where Sk-1 agrees with in its first k – 1 rows and the identity in its last n – k + 1 rows. Assume for simplicity that (this does not affect the final result). We have and Lemma 3.8 shows that the computed vector satisfies
which implies
and, from we Using (18.24) and exploiting have the form of (18.25) we find, after a little working, that
provided that To prove the last two parts of the theorem we exploit the Householder– MGS connection. By applying Theorem 18.4 to (18.19) we find that there is an orthogonal such that (18.26) with
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This does not directly yield (18.23), since is not orthonormal. However, it can be shown that if we define Q to be the nearest orthonormal matrix to in the Frobenius norm, then (18.23) holds with c3 = (see Problem 18.11). Now (18.21) and (18.23) yield
where c5 = c1 and we have used (18.23) to bound This bound implies (18.22) with c2 = 2c5 (use the first inequality in Problem 18.13). We note that (18.22) can be strengthened by replacing κ 2 (A) in the bound by the minimum over positive diagonal matrices D of κ 2 (AD). This follows from the observation that in the MGS method the computed Q is invariant under scalings A AD, at least if D comprises powers of the machine base. As a check, note that the bound in (18.18) for the case n = 2 is independent of the column scaling, since sin Theorem 18.12 tells us three things. First, the computed QR factors from the MGS method have a small residual. Second, the departure from orthonormality of Q is bounded by a multiple of κ 2 (A)u, so that Q is guaranteed to be nearly orthonormal if A is well conditioned. Finally, R is the exact triangular QR factor of a matrix near to A in a componentwise sense, so it is as good an R-factor as that produced by Householder QR factorization applied to A. In terms of the error analysis, the MGS method is weaker than Householder QR factorization only in that Q is not guaranteed to be nearly orthonormal. For the CGS method the residual bound (18.21) still holds, but no bound of the form (18.22) holds for n > 2 (see Problem 18.9). Here is a numerical example to illustrate the behaviour of the Gram– Schmidt methods. We take the 25 × 15 Vandermonde matrix A = where the pi are equally spaced on [0, 1]. The condition number κ 2 (A) = 1.47 × 109. We have
Both methods produce a small residual for the QR factorization. While CGS produces a Q showing no semblance of orthogonality, for MGS we have
18.8. Sensitivity of the QR Factorization How do the QR factors of a matrix behave under small perturbations of the matrix? This question was first considered by Stewart [944, 1977]. He showed
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that if
has rank n and
are QR factorization, then, for sufficiently small ∆A, (18.27) (18.28) where cn is a constant. Here, and throughout this section, we use the “economy size” QR factorization with R a square matrix normalized to have nonnegative diagonal elements. Similar normwise bounds are given by Stewart [951, 1993] and Sun [971, 1991], and, for AQ only, by Bhatia and Mukherjea [95, 1994] and Sun [975, 1995]. Componentwise sensitivity analyses have been given by Zha [1126, 1993] and Sun [972, 1992], [973, 1992]. Zha’s bounds can be summarized as follows, with the same assumptions and notation as for Stewart’s result above. Let |∆A| < where G is nonnegative with Then, for sufficiently small
(18.29) where c m,n is a constant depending on m and n. The quantity φ(A) = cond(R –1) can therefore be thought of as a condition number for the QR factorization under the columnwise class of perturbations considered. Note that φ is independent of the column scaling of A. As application of these bounds, consider a computed QR factorization A QR obtained via the Householder algorithm, where Q is the computed product of the computed Householder matrices. Theorem 18.4 shows that A+ ∆A = where is orthogonal and with ||G1 ||F = 1. Applying Lemma 18.3 to_ the computation of we have that where with ||G2 ||F = 1. From the expression we have, on applying (18.29) to the first term, (18.30) To illustrate this analysis we describe an example of Zha. Let
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where is a parameter. It is easy to verify that A = QRA and B = QRB are QR factorization (the same Q in each case) where
With
= 10–8, the computed Q factors from MATLAB are
and Since φ(A) 2.8 × 108 and φ(B) 3.8, the actual errors match the bound (18.30) very well. Note that κ 2 (A) 2.8 × 108 and κ2 (B) 2.1 × 108, so the normwise perturbation bound (18.28) is not strong enough to predict the difference in accuracy of the computed Q factors, unless we scale the factor out of the last column of B to leave a well-conditioned matrix. The virtue of the componentwise analysis is that it does not require a judicious scaling in order to yield useful results.
18.9. Notes and References The earliest appearance of Householder matrices is in the book by Turnbull and Aitken [1029, 1932, pp. 102–105]. These authors show that if ||x||2 = ||y||2 then a unitary matrix of the form R = azz* – I (in their notation) can be constructed so that Rx = y. They use this result to prove the existence of the Schur decomposition. The first systematic use of Householder matrices for computational purposes was by Householder [586, 1958], who used them to construct the QR factorization. Householder’s motivation was to compute the QR factorization with less arithmetic operations (in particular, less square roots) than are required by the use of Givens rotations. In the construction of §18.1 for a Householder matrix P such that Px = σe 1, the other choice of sign, sign(σ) = sign(x1), can be used, provided that υ 1 is formed in a numerically stable way. The appropriate formula is derived as follows [264, 1976], [819, 1971], [820, 1980, p. 91]:
A detailed analysis of different algorithms for constructing P such that Px = σe 1 is given by Parlett [819, 1971].
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Tsao [1023, 1975] describes an alternative way to form the product of a Householder matrix with a vector and gives an error analysis. There is no major advantage over the usual approach. As for Householder matrices, normwise error analysis for Givens rotations was given by Wilkinson [1087, 1963], [1089, 1965, pp. 131–139]. Wilkinson analysed QR factorization by Givens rotations for square matrices [1089, 1965, pp. 240-241], and his analysis was extended to rectangular matrices by Gentleman [435, 1973]. The idea of exploiting disjoint rotations in the error analysis was developed by Gentleman [436, 1975], who gave a normwise analysis that is simpler and produces smaller bounds than Wilkinson’s (our normwise bound in Theorem 18.9 is essentially the same as Gentleman’s). For more details of algorithmic and other aspects of Householder and Givens QR factorization, see Golub and Van Loan [470, 1989, §5.2]. The error analysis in §18.3 is a refined and improved version of analysis that appeared in the technical report [546, 1990] and was quoted without proof in Higham [549, 1991]. The WY representation for a product of Householder transformations should not be confused with a genuine block Householder transformation. Schreiber and Parlett [904, 1988] define, for a given (m > n), the “block reflector that reverses the range of Z” as
If n = 1 this is just a standard Householder transformation. A basic task is (m > n) find a block reflector H such that as follows: given
Schreiber and Parlett develop theory and algorithms for block reflectors, in both of which the polar decomposition plays a key role. Sun and Bischof [977, 1995] show that any orthogonal matrix can be expressed in the form Q = I – YSYT, even with S triangular, and they explore the properties of this representation. Another important use of Householder matrices, besides computation of the QR factorization, is to reduce a matrix to a simpler form prior to iterative computation of eigenvalues (Hessenberg or tridiagonal form) or singular values (bidiagonal form). For these two-sided transformations an analogue of Lemma 18.3 holds with normwise bounds (only) on the perturbation. Error analyses of two-sided application of Householder transformations is given by Ortega [811, 1963] and Wilkinson [1086, 1962], [1089, 1965, Chap. 6]. Mixed precision iterative refinement for solution of linear systems by Householder QR factorization is discussed by Wilkinson [1090, 1965, §10], who notes that convergence is obtained as long as a condition of the form cn κ 2 (A)u < 1 holds.
18.9 N OTES
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Fast Givens rotations can be applied to a matrix with half the number of multiplications of conventional Givens rotations, and they do not involve square roots. They were developed by Gentleman [435, 1973] and Hammarling [498, 1974]. Fast Givens rotations are as stable as conventional ones-see the error analysis by Parlett in [820, 19 8 0 , §6.8.3], for example-but, for the original formulations, careful monitoring is required to avoid overflow. Rath [861, 1982] investigates the use of fast Givens rotations for performing similarity transformations in solving the eigenproblem. Barlow and Ipsen [65, 1987] propose a class of scaled Givens rotations suitable for implementation on systolic arrays, and they give a detailed error analysis. Anda and Park [16, 1994] develop fast rotation algorithms that use dynamic scaling to avoid overflow. Rice [870, 1966] was the first to point out that the MGS method produces a more nearly orthonormal matrix than the CGS method in the presence of rounding errors. Björck [107, 1967] gives a detailed error analysis, proving (18.21) and (18.22) but not (18.23), which is an extension of the corresponding normwise result of Björck and Paige [119, 1992]. Björck and Paige give a detailed assessment of MGS versus Householder QR factorization. The difference between the CGS and MGS methods is indeed subtle. Wilkinson [1095, 1971] admitted that “I used the modified process for many years without even noticing explicitly that I was not performing the classical algorithm.” The orthonormality of the matrix from Gram-Schmidt can be improved by reorthogonalization, in which the orthogonalization step of the classical or modified method is iterated. Analyses of Gram–Schmidt with reorthogonalization are given by Abdelmalek [2, 1971], Ruhe [883, 1983], and Hoffmann [578, 1989]. Daniel, Gragg, Kaufman, and Stewart [263, 1976] analyse the use of classical Gram–Schmidt with reorthogonalization for updating a QR factorization after a rank one change to the matrix. The mathematical and numerical equivalence of the MGS method with Householder QR factorization of the matrix was known in the 1960s (see the Björck quotation at the start of the chapter) and the mathematical equivalence was pointed out by Lawson and Hanson [695, 1974, Ex. 19.39]. A block Gram-Schmidt method is developed by Jalby and Philippe [608, 1991 ] and error analysis given. See also Björck [115, 1994 ], who gives an up-to-date survey of numerical aspects of the Gram–Schmidt method. For more on Gram-Schmidt methods, including historical comments, see Björck [116, 1996]. One use of the QR factorization is to orthogonalize a matrix that, because of rounding or truncation errors, has lost its orthogonality; thus we compute A = QR and replace A by Q. An alternative approach is to replace (m > n) by the nearest orthonormal matrix, that is, the matrix Q that solves { ||A - Q|| : QTQ = I} = min. For the 2- and Frobenius norms, the optimal
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Q is the orthonormal polar factor U of A, where A = UH is a polar decomposition: has orthonormal columns and is symmetric positive semidefinite. If m = n, U is the nearest orthogonal matrix to A in any unitarily invariant norm, as shown by Fan and Hoffman [361, 1955]. Chandrasekaran and Ipsen [196, 1994] show that the QR and polar factors satisfy under the assumptions that A has full rank and columns of unit 2-norm and that R has positive diagonal elements. Sun [974, 1995] proves a similar result and also obtains a bound for ||Q — U||F in terms of ||ATA – I||F. Algorithms for maintaining orthogonality in long products of orthogonal matrices, which arise, for example, in subspace tracking problems in signal processing, are analysed by Edelman and Stewart [347, 1993] and Mathias [733, 1995]. Various iterative methods are available for computing the orthonormal polar factor U, and they can be competitive in cost with computation of a QR factorization. For more details on the theory and numerical methods, see Higham [530, 1986], [539, 1989], Higham and Papadimitriou [567, 1994], and the references therein. A notable omission from this chapter is a treatment of rank-revealing QR factorizations-ones in which the rank of A can readily be determined from R. This topic is not one where rounding errors play a major role, and hence it is outside the scope of this book. Pointers to the literature include Golub and Van Loan [470, 1989, §5.4], Chan and Hansen [192, 1992], and Björck [116, 199 6]. A column pivoting strategy for the QR factorization, described in Problem 18.5, ensures that if A has rank r then only the first r rows of R are nonzero. A perturbation theorem for the QR factorization with column pivoting is given by Higham [540, 1990]; it is closely related to the perturbation theory in §10.3.1 for the Cholesky factorization of a positive semidefinite matrix. 18.9.1. LAPACK LAPACK contains a rich selection of routines for computing and manipulating the QR factorization and its variants. Routine xGEQRF computes the QR factorization A = QR of an m × n matrix A by the Householder QR algorithm. If m < n (which we ruled out in our analysis, merely to simplify the notation), the factorization takes the form A = Q[ R 1 R2], where R1 is m × m upper triangular. The matrix Q is represented as a product of Householder transformations and is not formed explicitly. A routine xORGQR (or xUNGQR in the complex case) is provided to form all or part of Q, and routine xORMQR (or xUNMQR ) will pre- or postmultiply a matrix by Q or its (conjugate) transpose. Routine xGEQPF computes the QR factorization with column pivoting (see Problem 18.5). An LQ factorization is computed by xGELQF . When A is m × n with m < n
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387
it takes the form A = [L 0] Q. It is essentially the same as a QR factorization of AT and hence can be used to find the minimum 2-norm solution to an underdetermined system (see §20.1). LAPACK also computes two nonstandard factorization of an m × n A:
where L is lower trapezoidal and R upper trapezoidal.
Problems 18.1. Find the eigenvalues of a Householder matrix and a Givens matrix. 18.2. Let where and in Lemma 18.1. Derive a bound for
are the computed quantities described
18.3. A complex Householder matrix has the form
where and β = 2/υ∗υ. For given determine, if possible, P so that Px = y.
show how to
18.4. (Wilkinson [1089, 1965, p. 242]) Let and let P be a Householder matrix such that Px = ±||x||2 e1. Let G1,2, . . . . Gn– 1 ,n be Givens rotations such that Qx := G 1,2. . . Gn- 1 ,nx = ±||x||2 e1. True or false: P = Q? 18.5. In the QR factorization with column pivoting, columns are interchanged at the start of the kth stage of the reduction so that, in the notation of (18.1), ||xk|| 2 > ||Ck(:, j)||2 for all j. Show that the resulting R factor satisfies
so that, in particular, |r11| > |r22| > . . . > |rnn|. (These are the same equations as (10.13), which hold for the Cholesky factorization with complete pivoting—why?) 18.6. Let
be a product of disjoint Givens rotations. Show that
18.7. The CGS method and the MGS method applied to ( m > n) compute a QR factorization A = QR, Define the orthogonal projection Pi = where qi = Q(:, i). Show that
QR F ACTORIZATION
388
Show that the CGS method corresponds to the operations
while MGS corresponds to
18.8. Let A = [a 1, a2 ] θ, 0 < θ < π/2. (Thus, cosθ :=
18.9. (Björck [107,
1967])
and denote the angle between a 1 and a 2 by Show that
Let
which is a matrix of the form discussed by Lauchli [692, 196 1]. Assuming that evaluate the Q matrices produced by the CGS and MGS methods and assess their orthonormality. 18.10. Show that the matrix P in (18.19) has the form
where Q is the matrix obtained from the MGS method applied to A. 18.11. (Björck and Paige [119,
1992])
For any matrices satisfying
where both P11 and P21 have at least as many rows as columns, show that there exists an orthonormal Q such that A + ∆A = QR, where
(Hint: use the CS decomposition P11 = UCWT, P21 = VSWT, where U and V have orthonormal columns, W is orthogonal, and C and S are square, nonnegative diagonal matrices with C2 + S 2 = I. Let Q = VWT. Note, incident ally, that P21 = VWT · WSWT, so Q = VWT is the orthonormal polar factor of P21, and hence is the nearest orthonormal matrix to P21 in the 2- and Frobenius norms. For details of the CS decomposition see Golub and Van Loan [470, 1989, pp. 77, 471] and Paige and Wei [814, 1994].)
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18.12. We know that Householder QR factorization of is equivalent to the MGS method applied to A, and Problem 18.10 shows that the orthonormal matrix Q from MGS is a submatrix of the orthogonal matrix P from the Householder method. Since Householder’s method produces a nearly orthogonal P, does it not follow that MGS must also produce a nearly orthonormal Q? 18.13. (Higham [557, 1994]) Let position A = UH. Show that
(m > n) have the polar decom-
This result shows that the two measures of orthonormality ||ATA – I|| 2 ||A - U||2 are essentially equivalent (cf. (18.22)).
and
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Chapter 19
The Least Squares Problem
For some time it has been believed that orthogonalizing methods did not suffer this squaring of the condition number . . . It caused something of a shock, therefore, when in 1966 Golub and Wilkinson . . . asserted that already the multiplications QA and Qb may produce errors in the solution containing a factor x2(A). — A. VAN DER SLUIS, Stability of the Solutions of Linear Least Squares Problems (1975)
Most packaged regression problems do compute a cross-products matrix and solve the normal equations using a matrix inversion subroutine. All the programs . . . that disagreed (and some of those that agreed) with the unperturbed solution tried to solve the normal equations. — ALBERT E. BEATON, DONALD B. RUBIN, and JOHN L. BARONE, The Acceptability of Regression Solutions: Another Look at Computational Accuracy (1976)
On January 1, 1801 Giuseppe Piazzi discovered the asteroid Ceres. Ceres was only visible for forty days before it was lost to view behind the sun . . . Gauss, using three observations, extensive analysis, and the method of least squares, was able to determine the orbit with such accuracy that Ceres was easily found when it reappeared in late 1801. — DAVID K. KAHANER, CLEVE B. MOLER, and STEPHEN G. NASH, Numerical Methods and Software (1989)
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In this chapter we consider the least squares (LS) problem where A (m > n) has full rank. We begin by examining the sensitivity of the LS problem to perturbations. Then we examine the stability of methods for solving the LS problem, covering QR factorization methods, the normal equations and seminormal equations methods, and iterative refinement. Finally, we show how to compute the backward error of an approximate LS solution. We do not develop the basic theory of the LS problem, which can be found in standard textbooks (see, for example, Golub and Van Loan [470, 1989, However, we recall the fundamental result that any solution of the LS problem satisfies the normal equations ATAx = ATb (see Problem 19.1). Therefore if A has full rank there is a unique LS solution. More generally, + whatever the rank of A the vector xLS = A b is an LS solution, and it is the + solution of minimal 2-norm. Here, A is the pseudo-inverse of A (given by A + = (A T A) -l A T when A has full rank); see Problem 19.3. (For more on the pseudo-inverse see Stewart and Sun [954, 1990, 3.1]. )
19.1. Perturbation Theory Perturbation theory for the LS problem is, not surprisingly, more complicated than for linear systems, and there are several forms in which bounds can be stated. We begin with a normwise perturbation theorem that is a restatement of a result of Wedin [1069, 1973, Thm. 5.1]. For an arbitrary rectangular matrix A we define the condition number K2 (A) = ||A|| 2 ||A + || 2 . If A has r = rank(A) nonzero singular values, σ1 > · · · > σr, then K 2 (A) = σ1/σ r . Theorem 19.1 (Wedin). Let A rank, and let
Then, provided that
K2(A)ε
(m > n) and A + ∆A both be of full
< 1, (19.1) (19.2)
These bounds are approximately attainable.
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393
Proof. We defer a proof until 19.8, since the techniques used in the proof are not needed elsewhere in this chapter. The bound (19. 1) is usually interpreted as saying that the sensitivity of the LS problem is measured by k2 (A) when the residual is small or zero and by 2 K2(A) otherwise. This means that the sensitivity of the LS problem depends strongly on b as well as A, unlike for a square linear system. Here is a simple example where the K2(A)2 effect is seen:
It is a simple exercise to verify that
Since
K2(A)
= 1/ε,
Surprisingly, it is easier to derive componentwise perturbation bounds than normwise ones for the LS problem. The key idea is to express the LS solution and its residual as the solution of the augmented system (19.3) which is simply another way of writing the normal equations, ATAx = ATb. This is a square nonsingular system, so standard techniques can be applied. The perturbed system of interest is (19.4) where we assume that (19.5) From (19.3) and (19.4) we obtain
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Premultiplying by the inverse of the matrix on the left gives (19.6) Looking at the individual block components we obtain (19.7) (19.8) (Note that ||I – AA+||2 = min{1, m – n}, as is easily proved using the singular value decomposition (SVD).) On taking norms we obtain the desired perturbation result. Theorem 19.2. Let (m > n) and A + ∆A be of full rank. For the perturbed LS problem described by (19.4) and (19.5) we have (19.9) (19.10) for any monotonic norm. These bounds are approximately attainable. For a square system, we rem 7.4. Note, however, that s. For theoretical analysis it which x and r replace y and
have s = O, and we essentially recover Theothe bounds contain the perturbed vectors y and may be preferable to use alternative bounds in s and there is an extra factor
where the term in parentheses is assumed to be positive. For practical computation (19.9) is unsatisfactory because we do not know s = b + ∆ b – (A + ∆A)y. However, as Stewart and Sun observe [954, 1990, p. 159], = b – Ay is computable and and using this bound in (19.9) makes only a second-order change. The componentwise bounds enjoy better scaling properties than the normwise ones. If E = |A| and f = |b| then the bounds (19.7) and (19.8), and to a lesser extent (19.9) and (19.10), are invariant under column scalings (D diagonal). Row scaling does affect the componentwise bounds, since it changes the LS solution, but the componentwise bounds are less sensitive to the row scaling than the normwise bounds, in a way that is difficult to make precise.
19.2 S OLUTION
BY
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395
19.2. Solution by QR Factorization Let
with m > n and rank(A) = n. If A has the QR factorization
then
It follows that the unique LS solution is x =R -l c, and the residual ||b – Ax||2 = ||d||2 . Thus the LS problem can be solved with relatively little extra work beyond the computation of a QR factorization. Note that Q is not required explicitly; we just need the ability to apply Qt to a vector. It is well known that the Givens and Householder QR factorization algorithms provide a normwise backward stable way to solve the LS problem. The next result expresses this fact for the Householder method and also provides componentwise backward error bounds (essentially the same result holds for the Givens method). As in Chapter 18, we will use the generic constant γ cm, in which c denotes a small integer. We will assume implicitly that a condition holds of the form m nγ cm < 1/2. Theorem 19.3. Let A (m > n) have full rank and suppose the LS problem minx ||b – Ax||2 is solved using the Householder QR factorization method. The computed solution is the exact LS solution to
where the perturbations satisfy the normwise bounds
and the componentwise bounds
where ||G||F = 1. Proof. The proof is a straightforward generalization of the proof of Theorem 18.5 and is left as an exercise (Problem 19.2).
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As for Theorem 18.5 (see (18.7)), Theorem 19.3 remains true if we set ∆ b = 0, but in general there is no advantage to restricting the perturbations to A. Theorem 19.3 is a strong result, but it does not bound the residual of the computed solution, which, after all, is what we are trying to minimize. How close, then, is to minx ||b – Ax||2? We can answer this question using the perturbation theory of 19.1. With := b + ∆b – (A + ∆A) x := x LS and r := b – Ax, (19.6) yields
so that Substituting the bounds for ∆A and ∆b from Theorem 19.3, and noting that ||AA+||2 = 1, we obtain
where cond2(A) := || |A+||A| ||2. Hence
This bound contains two parts. The term mnγ cm || |b| + |A||x| ||2 is a multiple of the bound for the error in evaluating fl(b – Ax), and so is to be expected. The factor 1 + mnγ c m cond2 (AT) will be less than 1.1 (say) provided that cond2 (AT) is not too large. Note that cond2 (AT) < nk2 (A) and cond2 (AT) is invariant under column scaling of A (A A diag(d i), di 0). The conclusion is that, unless A is very ill conditioned, the residual b – will not exceed the larger of the true residual r = b – AX and a constant multiple of the error in evaluating fl(r)—a very satisfactory result.
19.3. Solution by the Modified Gram–Schmidt Method The modified Gram-Schmidt (MGS) method can be used to solve the LS problem. However, we must not compute x from x = R–1(QTb), because the lack of orthonormality of the computed would adversely affect the stability. Instead we apply MGS to the augmented matrix [A b]:
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397
We have
Since qn+l is orthogonal to the columns of Q1, ||b – Ax|| 22 = ||Rx – z|| 22 + ρ2 , so the LS solution is x = R-lZ. Of course, z = QT1b, but z is now computed as part of the MGS procedure instead of as a product between QT and b Björck [107, 1967] shows that this algorithm is forward stable, in the is as small as that for a normsense that the forward error ||x – wise backward stable method. It has recently been shown by Björck and Paige [119, 1992] that the algorithm is, in fact, normwise backward stable (see also Björck [115, 1994]), that is, a normwise result of the form in Theorem 19.3 holds. Moreover, a componentwise result of the form in Theorem 19.3 holds too—see Problem 19.5. Hence the possible lack of orthonormality of does not impair the stability of the MGS method as a means for solving the LS problem.
19.4. The Normal Equations The oldest method of solving the LS problem is to form and solve the normal equations, A T Ax = A T b. Assuming that A has full rank, we can use the following procedure: Form C = ATA and c = ATb. Compute the Cholesky factorization C = RTR. Solve RTy = c, Rx = y. Cost: : n 2(m + n/3) flops. If m >> n, the normal equations method requires about half as many flops as the Householder QR factorization approach (or the MGS method). However, it has less satisfactory numerical stability properties. There are two problems. The first is that information may be lost when = fl(ATA) is formed-essentially because forming the cross product is a squaring operation that increases the dynamic range of the data. A simple example is the matrix
for which
Even though A is distance approximately ε from a rank-deficient matrix, and hence unambiguously full rank if ε the computed cross product is
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singular. In general, whenever K2(A) > u –1/2 we can expect to be singular or indefinite, in which case Cholesky factorization is likely to break down (Theorem 10.7). The second weakness of the normal equations method is more subtle and is explained by a rounding error analysis. In place of C = ATA and c = ATb we compute
By Theorems 10.3 and 10.4, the computed Cholesky factor satisfy
and solution (19.11)
Overall, (19.12)
By bounding
with the aid of (19.11), we find that (19.13a) (19.13b)
These bounds show that we have solved the normal equations in a backward stable way, as long as ||A||2||b||2 ||ATb||2 . But if we try to translate this result into a backward error result for the LS problem itself, we find that the best backward error bound contains a factor K2(A) [569, 1987]. The best forward error bound we can expect, in view of (19.13), is of the form (19.14) (since K2 (A T A) = K2 (A)2). Now we know from Theorem 19.1 that the sensitivity of the LS problem is measured by K2(A)2 if the residual is large, but by K2(A) if the residual is small. It follows that the normal equations method has a forward error bound that can be much larger than that possessed by a backward stable method. A mitigating factor for the normal equations method is that, in view of Theorem 10.6, we can replace (19.14) by the (not entirely rigorous) bound
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where A = BD, with D = diag(||A(:, i)||2), so that B has columns of unit 2-norm. Van der Sluis’s result (Theorem 7.5) shows that
Hence the normal equations method is, to some extent, insensitive to poor column scaling of A. Although numerical analysts almost invariably solve the full rank LS problem by QR factorization, statisticians frequently use the normal equations (though perhaps less frequently than they used to, thanks to the influence of numerical analysts). The normal equations do have a useful role to play. In many statistical problems the regression matrix is contaminated by errors of measurement that are very large relative to the roundoff level; the effects of rounding errors are then likely to be insignificant compared with the effects of the measurement errors, especially if IEEE double precision (as opposed to single precision) arithmetic is used. The normal equations (NE) versus (Householder) QR factorization debate can be summed up as follows. ●
The two methods have a similar computational cost if m n, but the NE method is up to twice as fast for m >> n. (This statement assumes that A and b are dense; for details of the storage requirements and computational cost of each method for sparse matrices, see, for example, Björck [116, 1996] and Heath [510, 1984].)
●
The QR method is always backward stable. The NE method is guaranteed to be backward stable only if A is well conditioned.
●
The forward error from the NE method can be expected to exceed that for the QR method when A is ill conditioned and the residual of the LS problem is small.
●
The QR method lends itself to iterative refinement, as described in the next section. Iterative refinement can be applied to the NE method, but the rate of convergence inevitably depends on K2(A)2 instead of K2(A).
19.5. Iterative Refinement As for square linear systems, iterative refinement can be used to improve the accuracy and stability of an approximate LS solution. However, for the LS problem there is more than one way to construct an iterative refinement scheme. By direct analogy with the square system case, we might consider the scheme
T HE LEAST S QUARES P ROBLEM
400 1. Compute r = b – 2. Solve the LS problem mind ||Ad–r||2 . 3. Update y =
+ d.
(Repeat from step 1 if necessary, with
replaced by y).
This scheme is certainly efficient-a computed QR factorization (for example) of A can be reused at each step 2. Golub and Wilkinson [466, 1966] investigated this scheme and found that it works well only for nearly consistent systems. An alternative approach suggested by Björck [106, 1967] is to apply iterative refinement to the augmented system (19.3), so that both x and r are refined simultaneously. Since this is a square, nonsingular system, existing results on the convergence and stability of iterative refinement can be applied, and we would expect the scheme to work well. To make precise statements we need to examine the augmented system method in detail. For the refinement steps we need to consider an augmented system with an arbitrary right-hand side: r + Ax = f,
(19.15a)
ATr = g.
(19.15b)
If A has the QR factorization
where R
then (19. 15) transforms to
This system can be solved as follows: h = R - T g,
x = R –l (d l – h). The next result describes the effect of rounding errors on the solution process.
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Theorem 19.4. Let A be of full rank n < m and suppose the augmented system (19.15) is solved using a Householder QR factorization of A as described above. The computed and satisfy
where
with ||G||F = 1, ||Hi ||F = 1, i = 1:3. Proof. The proof does not involve any new ideas and is rather tedious, so we omit it. Consider first fixed precision iterative refinement. Theorem 19.4 implies to (19.15) satisfies that the computed solution
Unfortunately, this bound is not of a form that allows us to invoke Theorem 11.4. However, we can apply Theorem 11.3, which tells us that the corrected solution obtained after one step of iterative refinement satisfies
(19.16) Here, to the original bound (19.16). be included in and so be written
denotes the residual of the augmented system corresponding computed solution. We will make two simplifications to the = O(u), the first term in the bound may First, since 2 the O(u ) term. Second, (19.16) yields O(u) With these two simplifications, (19.16) may
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In view of the Oettli–Prager result (Theorem 7.3) this inequality tells us that, asymptotically, the solution produced after one step of fixed precision iterative refinement has a small componentwise relative backward error with respect to the augmented system. However, this backward error allows the two occurrences of A in the augmented system coefficient matrix to be perturbed differently, and thus is not a true componentwise backward error for the LS problem. Nevertheless, the result tells us that iterative refinement can be expected to produce some improvement in stability. Note that the bound in Theorem 19.2 continues to hold if we perturb the two occurrences of A in the augmented system differently. Therefore the bound is applicable to iterative refinement (with E = |A|, f = |b|), and so we can expect iterative refinement to mitigate the effects of poor row and column scaling of A. Numerical experiments show that these predictions are borne out in practice [549, 1991]. Turning to mixed precision iterative refinement, we would like to apply the analysis of $11.1, with “Ax = b“ again identified with the augmented system. However, the analysis of 11.1 requires a backward error result in which only the coefficient matrix is perturbed (see (11.1)). This causes no difficulties because from Theorem 19.4 we can deduce a normwise result (cf. Problem 7.7):
The theory of 11.1 then tells us that mixed precision iterative refinement will converge as long as the condition number of the augmented system matrix is not too large and that the rate of convergence depends on this condition number. How does the condition number of the augmented system relate to that of A? Consider the matrix that results from the scaling (α > 0): (19.17) Björck [106, 1967] shows that the eigenvalues of C(α) are
(19.18) ( 0’,
m - n times,
where σi i = 1: n, are the singular values of A, and that (19.19) with minα K2(C(α)) being achieved for α = (see Problem 19.7). Hence C (α) may be much more ill conditioned than A. However, in our analysis
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403
we are at liberty to take minα K2(C(α )) as the condition number, because scaling the LS problem according to b – Ax (b – Ax)/α does not change the computed solution or the rounding errors in any way (at least not if α is a power of the machine base). Therefore it is, K2(A) that governs the behaviour of mixed precision iterative refinement, irrespective of the size of the LS residual. As Björck [111, 1990] explains, this means that “in a sense iterative refinement is even more satisfactory for large residual least-squares problems.” He goes on to explain that “When residuals to the augmented system are accumulated in precision β -t2, t2 > 2t1, this scheme gives solutions to full single-precision accuracy even though the initial solution may have no correct significant figures.” Iterative refinement can be applied with the MGS method. Björck [106, 1967] gives the details and shows that mixed precision refinement works just as well as it does for Householder’s method.
19.6. The Seminormal Equations When we use a QR factorization to solve an LS problem minx ||b – Ax||2, the solution x is determined from the equation Rx = QTb (or via a similar expression involving Q for the MGS method). But if we need to solve for several right-hand sides that are not all available when the QR factorization is computed, we need to store Q before applying it to the right-hand sides. If A is large and sparse it is undesirable to store Q, as it can be much more dense than A. We can, however, rewrite the normal equations as R T Rx = A T b, which are called the seminormal equations. The solution x can be determined from these equations without the use of Q. Since the cross product matrix ATA is not formed explicitly and R is determined stably via a QR factorization, we might expect this approach to be more stable than the normal equations method. Björck [110, 1987] has done a detailed error analysis of the seminormal equations (SNE) method, under the assumption that R is computed by a backward stable method. His forward error bound for the SNE method is of the same form as that for the normal equations method, involving a factor 2 K2 (A) . Thus the SNE method is not backward stable. Björck considers applying one step of fixed precision iterative refinement to the SNE method, and he calls the resulting process the corrected seminormal equations (CSNE) method: R T Rx = A T b r = b – Ax
404
T HE LEAST S QUARES P ROBLEM R T Rw = A T r y=x+w
It is important that the normal equations residual be computed as shown, as T AT(b – Ax), and not as ATb – A Ax. Björck derives a forward error bound for the CSNE method that is roughly of the form
Hence, if k2(A)2U < 1, the CSNE method has a forward error bound similar to that for a backward stable method, and the bound is actually smaller than that for the QR method if k2(A)2U n), b and r = b – Ay. The normwise backward error (19.20) is given by
where
The backward error (19.20) is not a direct generalization of the usual normwise backward error for square linear systems, because it minimizes
19.7 B ACKWARD E RROR
405
Table 19.1. LS backward errors and residual for Vandermonde system.
instead of However, the Parameter θ allows us some flexibility: taking the limit forces ∆b = 0, giving the case where only A is perturbed. Theorem 19.5 can be interpreted as saying that if λ* > 0 then the backward error is essentially that given by Theorem 7.1 for a consistent system. If λ* < 0, however, the nearest perturbed system of which y is the LS solution is inconsistent. A sufficient condition for λ* < 0 is b range(A) (assuming µ 0), that is, the original system is inconsistent. The formulae given in Theorem 19.5 are unsuitable for computation because they can suffer from catastrophic cancellation when λ * < 0. Instead, the following alternative formula derived in [1060, 1995] should be used (see Problem 19.9): (19.21a) (19.21b) To illustrate Theorem 19.5, we consider an LS problem with a 25 x 15 Vandermonde matrix A = where the pi are equally spaced on [0, 1], and b = Ax with xi = i (giving a zero residual problem). The condition number k2 (A) = 1.47 x 109 . We solved the LS problem in MATLAB in four different ways: by using the NE with Cholesky factorization, via Householder QR factorization, and via the MGS method, using both the stable approach described in 19.3 and the unstable approach in which QTb is formed as a matrix–vector product (denoted MGS(bad)). The results, including the norms of the residuals r = b – are shown in Table 19.1. As would be expected from the analysis in this chapter, the QR and stable MGS methods produce backward stable solutions, but the NE method and the unstable MGS approach do not. As we saw in the analysis of iterative refinement, sometimes we need to consider the augmented system with different perturbations to A and AT. The next result shows that from the point of view of normwise perturbations
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T HE LEAST S QUARES P ROBLEM
and componentwise perturbations bounded by |∆ A| < εG|A|, the lack of “symmetry” in the perturbations has little effect on the backward error of y. Lemma 19.6 and Schwetlick). Let consider the perturbed augmented system
There is a vector
(m > n) and
and a perturbation ∆A with
∆ A = G1∆ A1 + G2 ∆ A 2 ,
(19.22)
such that
that is, y solves the LS problem minx ||(A + ∆A)x – b||2. Proof. If s = b– (A + ∆A1)y = 0 we take ∆A = ∆A1. Otherwise, we set ∆ A := P∆ A2 + (I – P)∆ A1 =: ∆A1 + PH, where P = ssT/sTs and H = ∆A2 – ∆A1. We have
where β = 1 – s T Hy/s T s. Then
Note that (19.22) implies a bound stronger than just ||∆ A||2 < ||∆ A1 ||2 + ||∆A 2 || 2 : p = 2, F. Turning to componentwise backward error, the simplest approach is to apply the componentwise backward error to the augmented system (19.3), setting
so as not to perturb the diagonal blocks I and O of the augmented system coefficient matrix. However, this approach allows A and AT to undergo different perturbations ∆A1 and ∆A2 with and thus does not give a true backward error, and Lemma 19.6 is of no help. This problem can be overcome by using a structured componentwise backward error to force symmetry
19.8 P ROOF
OF
W EDIN ’ S T HEOREM
407
of the perturbations; see Higham and Higham [527, 1992] for details. One problem remains: as far as the backward error of y is concerned, the vector r in the augmented system is a vector of free parameters, so to obtain the true componentwise backward error we have to minimize the structure-preserving componentwise backward error over all r. This is a nonlinear optimization problem to which no closed-form solution is known. Experiments show that when y is a computed LS solution, r = b – Ay is often a good approximation to the minimizing r [527, 1992], [549, 1991].
19.8. Proof of Wedin’s Theorem In this section we give a proof of Theorem 19.1. We define PA := AA+, the orthogonal projection onto range(A). Lemma 19.7. Let A, B B||2 < 1, then
If rank(A) = rank(B) and η = ||A+||2||A–
Proof. Let r = rank(A). A standard result on the perturbation of singular values gives that is, which gives the result on rearranging. Lemma 19.8. Let A, B
If rank(A) = rank(B) then
Proof. We have
The result then follows from the (nontrivial) equality ||PA(I – PB)||2 = ||PB(I– PA)||2; see, for example, Stewart [943, 1977, Thin. 2.3] or Stewart and Sun
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T HE LEAST S QUARES P ROBLEM
[954, 1990, Lem. 3.3.5], where proofs that use the CS decomposition are given. Proof of Theorem 19.1. Let B := A + AA. We have, with r = b – Ax, y – z = B+(b + ∆b) – z = B+(r + Ax + ∆b) – x = B + (r + Bx – ∆Ax + ∆b) – x = B + (r – ∆Ax + ∆b) – (I – B+B)x = B + ( r – ∆Ax + ∆b),
(19.23)
since B has full rank. Now B + r = B + (BB + )r = B + P B r = B + P B (I – P A )r.
(19.24)
Applying Lemmas 19.7 and 19.8, we obtain
(19.25) Similarly,
(19.26) The bound for ||x – y||2 /||x||2 is obtained by using inequalities (19.25) and (19.26) in (19.23). Turning to the residual, using (19.23) we find that s – r = ∆b + B(x – y) – ∆Ax = ∆b – BB+(r – ∆Ax + ∆b) – ∆Ax = (I – BB + )(∆b – ∆Ax) – BB+r. Since ||I – BB+||2 = min{l, m – n },
Using (19.24), Lemma 19.8, and ||BB+||2 = 1, we obtain
19.9 N OTES
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Hence
For the attainability, see Wedin [1069, 1973, 6]. Note that, as the proof has shown, Wedin’s theorem actually holds without any restriction on m and n, provided we define x = A + b and y = (A+ ∆A)+(b + ∆b) when m < n (in which case r = 0). We consider underdetermined systems in detail in the next chapter. The original version of Wedin’s theorem also requires only rank(A ) = rank(A + ∆A) and not that A have full rank.
19.9. Notes and References The most comprehensive and up to date treatment of the LS problem is the book by Björck [116, 1996], which is an updated and expanded version of [112, 1990]. It treats many aspects not considered here, including rank-deficient, weighted, and constrained problems. An early book devoted to numerical aspects of the LS problem was written by Lawson and Hanson [695, 1974], who, together with Stewart [94 1, 1973], were the first to present error analysis for the LS problem in textbook form. The history of the LS problem is described in the statistically oriented book by Farebrother [363, 1988]. The pseudo-inverse A+ underlies the theory of the LS problem, since the LS solution can be expressed as x = A+b. An excellent reference for perturbation theory of the pseudo-inverse is Stewart and Sun [954, 1990, 3.3]. The literature on pseudo-inverses is vast, as evidenced by the annotated bibliography of Nashed and Rail [786, 1976], which contains 1,776 references published up to 1976. Normwise perturbation theory for the LS problem was developed by various authors in the 1960s and 1970s. The earliest analysis was by Golub and Wilkinson [466, 1966], who gave a first-order bound and were the first to recognize the potential k2 (A)2 sensitivity. A nonasymptotic perturbation bound was given by Björck [107, 1967], who worked from the augmented system. An early set of numerical experiments on the Householder, Gram-Schmidt, and normal equations methods for solving the LS problem was presented by Jordan [618, 1968]; this paper illustrates the incomplete understanding of perturbation theory and error analysis for the LS problem at that time. van der Sluis [1041, 1975] presents a geometric approach to LS perturbation theory and gives lower bounds for the effect of worst-case perturbations. Golub and Van Loan [470, 1989, Thin. 5.3.1] give a first-order analogue of Theorem 19.1 expressed in terms of the angle θ between b and range(A ) instead of the residual r.
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T HE LEAST S QUARES P ROBLEM
Wei [1072, 1990] gives a normwise perturbation result for the LS problem with a rank deficient A that allows rank(A + ∆A) > rank(A). Componentwise perturbation bounds of the form in Theorem 19.2 were first derived by Björck in 1988 and variations have been given by Arioli, Duff, and de Rijk [25, 1989], Björck [113, 1991], and Higham [542, 1990]. Higham [542, 1990] examined the famous test problem from Longley [711, 1967]—a regression problem which has a notoriously ill-conditioned 16 x 7 coefficient matrix with k2 (A) 5 x 10 9 . The inequality (19.8) was found to give tight bounds for the effect of random componentwise relative perturbations of the problem generated in experiments of Beaton, Rubin, and Barone [86, 1976]. Thus componentwise perturbation bounds are potentially useful in regression analysis as an alternative to the existing statistically based techniques. The tools required for a direct proof of the normwise backward error result in Theorem 19.3 are developed in Wilkinson’s book The Algebraic Eigenvalue Problem [1089, 1965]. Results of this form were derived informally by Golub and Wilkinson (assuming the use of extended precision inner products) [466, 1966], stated by Wilkinson [1090, 1965, p. 93] and Stewart [941, 1973], and proved by Lawson and Hanson [695, 1974, Chap. 16]. The idea of using QR factorization to solve the LS problem was mentioned in passing by Householder [586, 1958]. Golub [463, 1965] worked out the details, using Householder QR factorization, and this method is sometimes called “Golub’s method”. In the same paper, Golub suggested the form of iterative refinement described at the start of 19.5 (which is implemented in a procedure by Businger and Golub [167, 1965]), and showed how to use QR factorization to solve an LS problem with a linear constraint Bx = c. It was Björck [106, 1967] who first recognized that iterative refinement should be applied to the augmented system for best results, and he gave a detailed rounding error analysis for the use of a QR factorization computed by the Householder or MGS methods. Björck and Golub [118, 1967] give an Algol code for computation and refinement of the solution to an LS problem with a linear constraint; they use Householder transformations, while Björck [108, 1968] gives a similar code baaed on the Gram–Schmidt method. In [109, 1978], Björck dispels some misconceptions of statisticians about (mixed precision) iterative refinement for the LS problem; he discusses standard refinement together with two versions of refinement based on the seminormal equations. Error analysis for solution of the LS problem by the classical GramSchmidt method with reorthogonalization is given by Abdelmalek [2, 1971], who obtains a forward error bound as good as that for a backward stable method. Higharn and Stewart [569, 1987] compare the normal equations method with the QR factorization method, with emphasis on aspects relevant to regression problems in statistics.
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Foster [398, 1991] proposes a class of methods for solving the LS problem that are intermediate between the normal equations method and the MGS method, and that can be viewed as block MGS algorithms. The most general analysis of QR factorization methods for solving the LS and related problems is by Björck and Paige [120, 1994], who consider an augmented system with an arbitrary right-hand side (see Problem 20.1) and prove a number of subtle stability results. Theorem 19.4 and the following analysis are from Higham [549, 1991]. Arioli, Duff, and de Rijk [25, 1989] investigate the application of fixed precision iterative refinement to large, sparse LS problems, taking the basic solver to be the block LDLT factorization code MA27 [329, 1982] from the Harwell Subroutine Library (applied to the augmented system); in particular, they use scaling of the form (19.17). Björck [114, 1992] determines, via an error analysis for solution of the augmented system by block LDLT factorization, a choice of α in (19.17) that minimizes a bound on the forward error. The idea of implementing iterative refinement with a precision that increases on each iteration (see the Notes and References to Chapter 11) can be applied to the LS problem; see Gluchowska and Smoktunowicz [453, 1990]. The use of SNE was first suggested by Kahan, in the context of iterative refinement, as explained by Golub and Wilkinson [466, 1966]. Stewart [945, 1977] discusses the problem of finding the normwise backward error for the LS problem and offers some backward perturbations that are candidates for being of minimal norm. The problem is also discussed by Higham [542, 1990]. Componentwise backward error for the LS problem has been investigated by Arioli, Duff, and de Rijk [25, 1989], Björck [113, 1991], and Higham [542, 1990]. Theorem 19.5 has been extended to the multiple right-hand side LS problem by Sun [976, 1996]. Lemma 19.6 is from a book by and Schwetlick, which has been published in German [658, 1988] and Polish [659, 1992] editions, but not in English. The lemma is their Lemma 8.2.11, and can be shown to be equivalent to a result of Stewart [943, 1977, Thin. 5.3]. Other methods for solving the LS problem not considered in this chapter include those of Peters and Wilkinson [826, 1970], Cline [215, 1973], and Plemmons [835, 1974], all of which begin by computing an LU factorization of the rectangular matrix A. Error bounds for these methods can be derived using results from this chapter and Chapters 9 and 18. In this chapter we have not specifically treated LS problems whose coefficient matrices have rows varying greatly in norm, and we have not considered weighted LS problems minx ||D(b — Ax)||2, where D = diag(d i ). Error analysis for Householder QR factorization with column pivoting applied to badly row-scaled problems is given by Powell and Reid [840, 1969]. Methods and error analysis for weighted LS problems are given by Barlow [59, 1988], Bar-
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T HE LEAST S QUARES P ROBLEM
low and Handy [64, 1988], Barlow and Vemulapati [66, 1992], Gulliksson and Wedin [489, 1992], Gulliksson [487, 1994], [488, 1995], and Van Loan [1042, 1985]. Another topic not considered here is the constrained LS problem, where x is required to satisfy linear equality and/or inequality constraints. Numerical methods are described by Lötstedt [713, 1984] and Golub and Van Loan [470, 1989, 12. 1], and perturbation theory is developed by Eldén [350, 1980], Lötstedt [712, 1983], and Wedin [1070, 1985].
19.9.1. LAPACK Driver routine xGELS solves the full rank LS problem by Householder QR factorization. It caters for multiple right-hand sides, each of which defines a separate LS problem. Thus, xGELS solves min{ ||B – AX||F : X }, where A This routine does not return any error bounds, and iterative refinement is not supported for LS problems in LAPACK. Driver routines xGELSX and xGELSS solve the rank-deficient LS problem with multiple right-hand sides, using, respectively, a complete orthogonal factorization (computed via QR factorization with column pivoting) and the SVD. LAPACK also contains routines for solving the linearly constrained LS problem (xGGLSE) and a generalized form of weighted LS problem (xGGGLM).
Problems 19.1. Show that any solution to the LS problem minx ||b – Ax||2 satisfies the normal equations A T Ax = A T b. What is the geometrical interpretation of these equations? 19.2. Prove Theorem 19.3. 19.3. The pseudo-inverse X can be defined as the unique matrix satisfying the four Moore–Penrose conditions (i) AXA = A, (iii) AX = (AX)T,
(ii) XAX = X, (iv) XA = (XA) T .
Let A = UΣ VT be an SVD, with Σ = diag(σi ) and let r = rank(A ). Show that X = V diag(σ1-l, . . . ,σr-l. 0, . . . , 0)UT satisfies (i)-(iv) and hence is the pseudo-inverse of A. Show that (A+)+ = A. 19.4. Show that the pseudo-inverse A+ of A
solves the problem
P ROBLEMS
413
Is the solution unique? 19.5. Prove a result analogous to Theorem 19.3 for the MGS method, as described in 19.3. 19.6. Consider the LS problem minx||b – Ax||2, where Let be the computed LS solution obtained from the normal equations method and x the exact solution, and define Using (19.12) and (19.13) show that a bound holds of the form
19.7. Prove (19.18) and (19.19). 19.8. (Waldén, Karlson, and Sun [1060, 1995]) Partially complete the gap in Theorem 19.5 by evaluating η F (0) for the case that is, ∆b = 0. 19.9. Prove (19.21).
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Chapter 20
Underdetermined Systems
I’m thinking of two numbers. Their average is 3. What are the numbers? — CLEVE B. MOLER, The World’s Simplest Impossible Problem (1990)
This problem arises in important algorithms used in mathematical programming . . . In these cases, B is usually very large and sparse and, because of storage difficulties, it is often uneconomical to store and access Q1 . . . Sometimes it has been thought that [the seminormal equations method] could be disastrously worse than [the Q method] . . . It is the purpose of this note to show that such algorithms are numerically quite satisfactory. — C. C. PAIGE, An Error Analysis of a Method for Solving Matrix Equations (1973)
415
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U NDERDETERMINED S YSTEMS
Having considered well-determined and overdetermined linear systems, we now turn to the remaining class of linear systems: those that are underdetermined.
20.1. Solution Methods Consider the underdetermined system Ax = b, where A The system can be analysed using a QR factorization
with m < n.
(20.1) where Q is orthogonal and R is upper triangular. (We could, alternatively, use an LQ factorization of A, but we will keep to the standard notation.) We have b = Ax = [RT 0]Q T x = R T y 1 ,
(20.2)
where
If A has full rank then yl = R-Tb is uniquely determined and all solutions of Ax = b are given by
The unique solution xLS that minimizes ||x||2 is obtained by setting y2 = 0. We have (20.3)
(20.4)
where A + = A T (AA T ) – 1 is the pseudo-inverse of A. Hence x L S can be characterized as xLS = ATy, where y solves the normal equations AATy = b. Equation (20.3) defines one way to compute xLS . We will refer to this method as the “Q method”. When A is large and sparse it is desirable to avoid storing and accessing Q, which can be expensive. An alternative method
20.2 P ERTURBATION T HEORY
417
with this property uses the QR factorization (20.1) but computes x LS as x LS = A T y, where (20.5) RTRy = b (cf. (20.4)). These latter equations are called the seminormal equations (SNE). As the “semi” denotes, however, this method does not explicitly form AAT, which would be undesirable from the standpoint of numerical stability. Note that equations (20.5) are different from the equations RTRx = ATb for an overdetermined least squares (LS) problem, where A = Q[ RT 0] T with m > n, which are also called seminormal equations (see §19.6).
20.2. Perturbation Theory A componentwise perturbation result for the minimum 2-norm solution to an underdetermined system is readily obtained. Theorem 20.1 (Demmel and Higham). Let A rank and Suppose ||A+ ∆A||2 < 1 and
(m < n) be of full
If x and y are the minimum 2-norm solutions to Ax = b and (A + ∆A)y = b + ∆b, respectively, then, for any monotonic norm,
(20.6) For any Hölder p-norm, the bound is attainable to within a constant factor depending on n. Proof. The perturbed matrix A + ∆A = A(I + A+ ∆A) has full rank, so we can manipulate the equation y = (A+ ∆A)T((A + ∆A)(A + ∆A)T)-l(b + ∆b) to obtain y – x = (I – A+ A) ∆AT(AAT)-lb + A+(∆b – ∆Ax) + O( 2 ) = (I - A + A) ∆ATA+Tx + A+(∆b – ∆Ax) + O(ε2).
(20.7)
The required bound follows on using absolute value inequalities and taking norms. That the bound is attained to within a constant factor depending on n for Holder p-norms is a consequence of the fact that the two vectors on the right-hand side of (20.7) are orthogonal.
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U NDERDETERMINED S YSTEMS
Two special cases are worth noting, for later use. We will use the equality ||I - A+A||2 = min{l, n - m}, which can be derived by consideration of the QR factorization (20.1), for example. If E = |A|H, where H is a given nonnegative matrix, and f = |b|, then we can put (20.6) in the form (20.8) where cond2 ( A) = || |A+||A| ||2. Note that cond2 (A ) is independent of the row scaling of A (cond2(D A) = cond 2 (A) for nonsingular diagonal D). If E = ||A||2emeTn and f = ||b||2em, where em denotes the m-dimensional vector of 1s, then (20.9) The following analogue of Lemma 19.6 will be needed for the error analysis in the next section, It says that if we perturb the two occurrences of A in the normal equations AATx = b differently, then the solution of the perturbed system is the solution of normal equations in which there is only one perturbation of A and, moreover, this single perturbation is no larger, in the normwise or componentwise sense, than the two perturbations we started with. Lemma 20.2 full rank and suppose (A + ∆A 1 )
and Schwetlick). Let A
= b,
(m < n) be of
= (A + ∆A 2 ) T
Assume that 3 max (||Α+∆A1||2, ||A+ ∆A2||2) < 1. Then there is a vector and a perturbation ∆A with ∆A = ∆AIGI + ∆A 2 G 2 ,
Gl + G2 = I,
such that that is,
is the minimum 2-norm solution to (A + ∆A)x = b.
Proof. The proof is similar to that of Lemma 19.6, but differs in some details. If = (A+ ∆A 2 ) T = 0 we take ∆A = ∆A2. otherwise, we set ∆ A := ∆AIP + ∆A2(I – P) =: ∆A2 + HP,
20.3 E RROR ANALYSIS where
419
and H = ∆A1 - ∆A2. We have
where β = which shows that we need to set check that (A + ∆A ) = b, we evaluate
as required. The vector
To
is undefined if β = 0. But
which is positive if 3 max (||A+ ∆A 1 || 2 , ||A + ∆A 2 || 2 ) < 1. Note that in Lemma 20.2 we have the normwise bound
20.3. Error Analysis We now consider the stability of the Q method and the SNE method. For both methods we assume that the QR factorization is computed using Householder or Givens transformations. Before presenting the results we define a measure of stability. The componentwise backward error for a minimum-norm underdetermined system Ax = b is defined as s.t. y is the min. norm solution to (A+ ∆A)y = b + ∆b }, Note the requirement in this definition that y be the minimum norm solution; the usual componentwise backward error (see (7.6)) is a generally smaller quantity. Let us say that a method is row-wise backward stable if it produces a computed solution for which the componentwise backward error
420
U NDERDETERMINED S YSTEMS
is of order u, where E = |A|emeTn and f = |b|. This condition requires that solve a perturbed minimum norm problem in which the perturbations to the ith row of A are small compared with the norm of that row (similarly for b); cf. the discussion of componentwise backward errors in 7.2. Theorem 20.3. Let A with rank(A) = m < n, and assume that a rendition of the form cond2 (A)mnγ c n < 1 holds. Suppose the underdetermined system AX = b is solved in the minimum 2-norm sense using the Q method. Then the computed solution is the minimum 2-norm solution to (A+ ∆A)x = b, where and ||G||F = 1. Proof. The Q method solves the triangular system R T y l = b and then forms x = Assuming the use of Householder QR factorization, from Theorem 18.4 we have that
for some orthogonal matrix Q, where ||∆A0||F < mγ cn||A||F with ||G0 ||F = 1. The computed satisfies
From Lemma 18.3, the computed solution
and
satisfies (20.10)
We now rewrite the latter two equations in such a way that we can apply Lemma 20.2:
It is straightforward to show that |∆ Ai | < mnγ´cn|A|Gi , and ||∆Ai ||F < Lemma 20.2.
mγ´||A||F, i = 1:2.
||Gi ||F = 1, i = 1:2, The result follows on invocation of
20.3 E RROR ANALYSIS
421
Theorem 20.3 says that the Q method is row-wise backward stable. This is not altogether surprising, since (Householder or Givens) QR factorization for the LS problem enjoys an analogous backward stability result (Theorem 19.3), albeit without the restriction of a minimum norm solution. Applying (20.8) to Theorem 20.3 we obtain the forward error bound (20.11) The same form of forward error bound (20.11) can be derived for the SNE method as for the Q method [292, 1993]. However, it is not possible to obtain a result analogous to Theorem 20.3, nor even to obtain a residual bound of the form (which would imply that solved a nearby system, though would not necessarily be the minimum norm solution). The method of solution guarantees only that the seminormal equations themselves have a small residual. Thus, as in the context of overdetermined LS problems, the SNE method is not backward stable. A possible way to improve the stability is by iterative refinement, as shown in [292, 1993]. Note that the forward error bound (20. 11) is independent of the row scaling of A, since cond2 (A) is. The bound is therefore potentially much smaller than the bound
obtained by Paige [813, 1973] for the SNE method and by Jennings and Osborne [614, 1974] and Arioli and Laratta [26, 1985, Thin. 4] for the Q method. Finally, we mention an alternative version of the Q method that is based on the modified Gram–Schmidt (MGS) method. The obvious approach is to compute the QR factorization AT = QR using MGS solve RTy = b, and then form x = Qy. Since Q is provided explicitly by the MGS method, the final stage is a full matrix–vector multiplication, unlike for the Householder method. However, because the computed Q may depart from orthonormality, this method is unstable in the form described. The formation of x = Qy should instead be done as follows:
The recurrence can be written as x(k-l) = x(k) + ykq k – (qkTx(k))qk, and the last term is zero in exact arithmetic if the qk are mutually orthogonal. In finite precision arithmetic this correction term has the “magical” effect of making
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Table 20.1. Backward errors for underdetermzned Vandermonde system.
Householder QR MGS with x := Qy MGS with x formed stably (see text) SNE method (using Householder QR)
9.76 X 10-18 4.10 x 10-4 2.25 X 10-17 1.99 x 10-4
the algorithm stable, in the sense that it satisfies essentially the same result as the Q method in Theorem 20.3; see Björck and Paige [120, 1994]. A numerical example is instructive. Take the 20 x 30 Vandermonde matrix where the pi are equally spaced on [0, 1], and let b have elements equally spaced on [0, 1]. The condition number k2 (A) = 4.35 x 1014 . The (standard) normwise backward errors in the 2-norm are shown in Table 20.1. For A T , the supplied by MGS satisfies = 1.41 x 10-3, which explains the instability of the “obvious” MGS solver.
20.4. Notes and References The seminormal equations method was suggested by Gill and Murray [442, 1973] and Saunders [894, 1972]. Other methods for obtaining minimal 2-norm solutions of underdetermined systems are surveyed by Cline and Plemmons [219, 1976]. Theorem 20.1 is from Demmel and Higham [292, 1993]. The bound (20.9) is well known; it follows from Wedin’s original version of our Theorem 19.1, which applies to minimum 2-norm underdetermined problems as well as LS problems. Theorem 20.3 is new. Demmel and Higham [292, 1993] prove the weaker result that from the Q method is very close to a vector that satisfies the criterion for row-wise backward stability, and Lawson and Hanson [695, 1974, Thin. 16. 18] give a corresponding result in which satisfies the criterion for general normwise backward stability. The key to showing actual backward stability is the use of and Schwetlick’s lemma, which is a modification of Lemma 8.2.11 in [658, 1988] and [659, 1992] (our Lemma 19.6). Demmel and Higham [292, 1993] also give error analysis for the seminormal equations method. The new MGS algorithm for solving the minimum norm problem was first suggested by Björck and Paige [119, 1992]; see also Björck [115, 1994]. Arioli and Laratta [27, 1986] give error analysis of QR factorization methods for solving the general problem min{ ||x – c||2 : Ax = b }, where A
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with m < n.
20.4.1. LAPACK The same routines that solve the (overdetermined) LS problem also solve underdetermined systems for the solution of minimal 2-norm. Thus, xGELS solves a full rank underdetermined system with multiple right-hand sides by the Q method. Routines xGELSX and xGELSS solve rank-deficient problems with multiple right-hand sides, using, respectively, a complete orthogonal factorization (computed via QR factorization with column pivoting) and the singular value decomposition.
Problems 20.1. (Björck [114, 1992]) Show that the system (20.12) characterizes the solution to the following generalizations of the LS problem and the problem of finding the minimum norm solution to an underdetermined system: (20.13) (20.14) 20.2. (RESEARCH PROBLEM) Find a formula for the backward error of an arbitrary approximation to the minimum 2-norm solution of an underdetermined system. That is, for A with rank(A) = m < n, find m(y) := min{ e : y is the minimum 2-norm solution to (A + ∆A)y = b + ∆b, where ||∆A||2 < e||A||2, ||∆b||2 < e||b||2 }.
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Chapter 21
Vandermonde Systems
We began, 25 years ago, the The original motivation came from computation of Gauss moments of
to take up [the conditioning of] class of Vandermonde matrices. unpleasant experiences with the type quadrature rules from the the underlying weight function.
— WALTER GAUTSCHI, How (Un)stable are Vandermonde Systems? (1990)
Extreme ill-conditioning of the [Vandermonde] linear systems will eventually manifest itself as n increases by yielding an error curve which is not sufficiently Ievelled on the current reference . . . or more seriously fails to have the correct number of sign changes. — M. ALMACANY, C. B. DUNHAM, and J. WILLIAMS, Discrete Chebyshev Approximation by Interpolating Rationals (1984)
425
V ANDERMONDE S YSTEMS
426
A Vandermonde matrix is defined in terms of scalars α0, α1, . . . . αn
by
Vandermonde matrices play an important role in various problems, such as in polynomial interpolation. Suppose we wish to find the polynomial pn(x) = anxn + an-1xn–1 + · · · + a0 that interpolates to the data for distinct points αi , that is, pn(α i ) = fi , i = 0:n. Then the desired coefficient vector a = [a0, a1, . . . ,an ] T is the solution of the dual Vandermonde system V T a = f (dual). The primal system Vx = b (primal) represents a moment problem, which arises, for example, when determining the weights for a quadrature rule: given moments bi find weights xi such that Because a Vandermonde matrix depends on only n+ 1 parameters and has a great deal of structure, it is possible to perform standard computations with reduced complexity. The easiest algorithm to derive is for matrix inversion.
21.1. Matrix Inversion Assume that V is nonsingular and let V the equation WV = I may be written
-1
= W =
The ith row of
These equations specify a fundamental interpolation problem that is solved by the Lagrange basis polynomial: (21.1)
The inversion problem is now reduced to finding the coefficients of li (x). It is clear from (21.1) that V is nonsingular iff the αi are distinct. It also follows from (21. 1) that V -1 is given explicitly by
427
21.1 M ATRIX I NVERSION
where σk(y1, . . . , yn) denotes the sum of all distinct products of k of the arguments y1 , . . . , yn (that is σ k is the kth elementary symmetric function). An efficient way to find the wij is first to form the master polynomial
and then to recover each Lagrange polynomial by synthetic division:
The scalars qi (α i ) can be computed by Homer’s rule as the coefficients of qi are formed. Algorithm 21.1. Given distinct scalars α0, α1, . . . . ,αn computes W =
this algorithm
% Stage 1: Construct the master polynomial. a0 = −α0; a1 = 1 for k = l:n ak+l = 1 for j = k: –1: 1 a j = a j – 1 - αk a j end
% Stage 2: Synthetic division. for i = 0:n win = l; s = l for j = n – 1: –1:0 w i j = a j + l + αi w i , j + l s = αi s + wij end w(i, : ) = w(i, : )/s end Cost: 6n2 flops. The O(n2) complexity is optimal, since the algorithm has n2 output values, each of which must partake in at least one operation. Vandermonde matrices have the deserved reputation of being ill conditioned. The ill conditioning is a consequence of the monomials being a poor
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Table 21.1. Bounds and estimates for αi (VI): (V2): (V3): (V4): (V5):
1/i+l arbitrary αi > 0 equispaced [0,1] equispaced [– 1, 1]
Reference [428, 1990] [1035, 1994] [429, 1988] [428, 1990] [425, 1975]
(V6): (V7):
Chebyshev nodes [–1, 1] roots of unity
[425, 1975] well known.
Bound or estimate
basis for the polynomials on the real line. A variety of bounds for the condition number of a Vandermonde matrix have been derived by Gautschi and his co-authors. Let Vn = V(α0, α1, . . . , α n–1 ) For arbitrary distinct points αi , (21.3)
with equality on the right when for all j with a fixed θ (in particular, when αj > 0 for all j) [424, 1962], [426, 1978]. Note that the upper and lower bounds differ by at most a factor 2 n–1. More specific bounds are given in Table 21.1, on which we now comment. Bound (V1) and estimate (V4) follow from (21.3). The condition number for the harmonic points 1/(i + 1) grows faster than n!; by contrast, the condition numbers of the notoriously ill-conditioned Hilbert and Pascal matrices grow only exponentially (see 26.1 and 26.4). For any choice of points the rate of growth is at least exponential (V2), and this rate is achieved for points equally spaced on [0, 1]. For points equally spaced on [– 1, 1], the condition number grows at a slower exponential rate than that for [0, 1], and the growth rate is slower still for the zeros of the nth degree Chebyshev polynomial (V6). For one set of points the Vandermonde matrix is perfectly conditioned: the roots of unity, for which is unitary.
21.2. Primal and Dual Systems The standard Vandermonde matrix can be generalized in at least two ways: by allowing confluence of the points αi and by replacing the monomials by
21.2 P RIMAL
AND
D UAL S YSTEMS
429
other polynomials. An example of a confluent Vandermonde matrix is
(21.4)
The second, third, and fifth columns are obtained by “differentiating” the previous column. The transpose of a confluent Vandermonde matrix arises in Hermite interpolation; it is nonsingular if the points corresponding to the “nonconfluent columns” are distinct. A Vandermonde-like matrix is defined by where pi is a polynomial of degree i. The case of practical interest is where the pi satisfy a three-term recurrence relation. In the rest of this chapter we will assume that the pi do satisfy a three-term recurrence relation. A particular application is the solution of certain discrete Chebyshev approximation problems [11, 1984]. Incorporating confluence, we obtain a confluent Vandermonde-like matrix, defined by P = P(α0, α1,. . . ,αn ) = [q0 (α0), q1 (α1), . . . . qn (α n )] where the αi are ordered so that equal points are contiguous, that is, (21.5) and the vectors qj(x) are defined recursively by if j = 0 or otherwise. For all polynomials and points, P is nonsingular; this follows from the derivation of the algorithms below. One reason for the interest in Vandermonde-like matrices is that for certain polynomials they tend to be better conditioned than Vandermonde matrices (see, for example, Problem 21.5). Gautschi [427, 1983] derives bounds for the condition numbers of Vandermonde-like matrices. Fast algorithms for solving the confluent Vandermonde-like primal and dual systems Px = b and PTa = f can be derived under the assumption that the p j(x) satisfy the three-term recurrence relation (21.6a) (21.6b) where for all j. Note that in this chapter γi denotes a constant in the recurrence relation and not iu/(1 – iu) as elsewhere in the book. The latter notation is not used in this chapter.
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430
The algorithms exploit the connection with interpolation. Denote by r(i) > 0 the smallest integer for which αi = αi-1 = · · · = αr(i). Considering first the dual system PTa = f, we note that (21.7) satisfies i = 0:n. Thus ψ is a Hermite interpolating polynomial for the data {αi, fi }, and our task is to obtain its representation in terms of the basis As a first step we construct the divided difference form of ψ, (21.8) The (confluent) divided differences ci = f[α0, α1,. . . . αi] may be generated using the recurrence relation
(21.9) Now we need to generate the ai in (21.7) from the ci in (21.8). The idea is to expand (21.8) using nested multiplication and use the recurrence relations (21.6) to express the results as a linear combination of the pj. Define q n (x) = cn ,
(21.10) (21.11)
from which q0(x) = ψ(x). Let (21.12)
To obtain recurrences for the coefficients of (21.11), giving
we expand the right-hand side
21.2 P RIMAL
AND
D UAL SYSTEMS
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Using the relations, from (21.6),
we obtain, for k = 0:n – 2,
(21.13) in which the empty summation is defined to be zero. For the special case k = n – l we have (21.14) Recurrences for the coefficients in terms of j = k +1: n, follow immediately by comparing (21.12) with (21.13) and (21.14). In the following algorithm, stage I computes the confluent divided differences and stage II implements the recurrences derived above. Algorithm 21.2 (dual, PTa = f). Given parameters a vecsatisfying (21 .5), this algorithm solves the tor f, and points dual confluent Vandermonde-like system PTa = f. % Stage I: Set c = f
V ANDERMONDE S YSTEMS
432 for k = 0:n – 1 clast = ck for j = k + l:n i f αj αj-k-1 then c j = c j/(k + 1) else temp = cj c j = (cj – clast)/(αj – αj – k – 1 ) clast = temp end end end % Stage II: Set a = c a n - l = a n - l + (β 0 - αn - 1 )a n a n = a n /θ 0 for k = n – 2: – 1:0 for j = l:n – k –2 end a n = a n /θ n - k - 1 end
Assuming that the values are given (note that γj appears only in the terms the computational cost of Algorithm 21.2 is at most 9n 2 / 2 flops. The vectors c and a have been used for clarity; in fact both can be replaced by f, so that the right-hand side is transformed into the solution without using any extra storage. Values of for some polynomials of interest are collected in Table 21.2. The key to deriving a corresponding algorithm for solving the primal system is to recognize that Algorithm 21.2 implicitly computes a factorization of P - T into the product of 2n triangular matrices. In the rest of this chapter we adopt the convention that the subscripts of all vectors and matrices run from 0 to n. In stage I, letting c(k) denote the vector c at the start of the k t h iteration of the outer loop, we have c(0) = f,
c(k+l) = Lkc(k),
k = 0:n – 1.
(21.15)
The matrix Lk is lower triangular and agrees with the identity matrix in rows
21.2 P RIMAL
AND
433
D UAL SYSTEMS
Table 21.2. Parameters in the three-term recurrence (21.6). Polynomial Monomials Chebyshev Legendre* Hermite Laguerre
θj 1 2* 2
βj 0 0 0 0 2j + l
γj 0 1
∗
*
θ0 = 1 p j (1) = 1
2j
0 to k. The remaining rows can redescribed, for k + l < j < n, by if αj = αj - k - 1 , some s < j, otherwise, where ej is column j of the identity matrix. Similarly, stage II can be expressed as a (n) = c (n) , a ( k ) = U k a ( k + l ) , k = n – 1: – l:0. (21.16) The matrix Uk is upper triangular, it agrees with the identity matrix in rows 0 to k – 1, and it has zeros everywhere above the first two superdiagonals. From (21.15) and (21.16) we see that the overall effect of Algorithm 21.2 is to evaluate, step-by-step, the product a = U 0 . . .U n-l L n-l . . . L 0 f = P-T f.
(21.17)
Taking the transpose of this product we obtain a representation of P-1, from which it is easy to write down an algorithm for computing x = P–1b. Algorithm 21.3 (primal, Px = b). Given parameters a vecsatisfying (21.5), this algorithm solves the tor b, and points primal confluent Vandermonde-like system Px = b. l % Stage I: Set d = b for k = 0:n – 2 for j = n – k: – 1:2 end end
434
V ANDERMONDE S YSTEMS % Stage II: Set x = d for k = n – 1: – 1:0 xlast = 0 for j = n: - l :k + l i f αj = αj–k–l then x j = x j/(k + 1) else t e m p = x j/(α j - αj - k - 1 ) xj = temp – xlast xlast = temp end end xk = xk - xlast end
Algorithm 21.3 has, by construction, the same operation count as Algorithm 21.2.
21.3. Stability Algorithms 21.2 and 21.3 have interesting stability properties. Depending on the problem parameters, the algorithms can range from being very stable (in either a backward or forward sense) to very unstable. When the pa are the monomials and the points αi are distinct, the algorithms reduce to those of Björck and Pereyra [121, 1970]. Björck and Pereyra found that for the system Vx = b with αi = 1/(i + 3), bi = 2-i, n = 9, and on a computer with u 10 –16 ,
Thus the computed solution has a tiny componentwise relative error, despite the extreme ill condition of V. Björck and Pereyra comment “It seems as if at least some problems connected with Vandermonde systems, which traditionally have been considered too ill-conditioned to be attacked, actually can be solved with good precision.” This high accuracy can be explained with the aid of the error analysis below. The analysis can be kept quite short by exploiting the interpretation of the algorithms in terms of matrix–vector products. Because of the inherent duality between Algorithms 21.2 and 21.3, any result for one has an analogue for the other, so we will consider only Algorithm 21.2. The analysis that follows is based on the model (2.4), and so is valid only for machines with a guard digit. With the no-guard-digit model (2.6) the
21.3 S TABILITY
435
bounds become weaker and more complicated, because of the importance of terms in the analysis. 21.3.1. Forward Error Theorem 21.4. If no underflow or overflows are encountered then Algorithm 21.2 runs to completion and the computed solution satisfies (21.18) where c(n, u) := (1 + u)7n – 1 = 7nu + 0(u2). Proof. First, note that Algorithm 21.2 must succeed in the absence of underflow and overflow, because division by zero cannot occur. The analysis of the computation of the c(k) vectors is exactly the same as that for the nonconfluent divided differences in 5.3 (see (5.9) and (5.10)). However, we obtain a slightly cleaner error bound by dropping the γ k notation and instead writing |∆Lk| < [(1 + u)3 - 1]|Lk|.
(21.19)
Turning to the equations (21.16), we can regard the multiplication a(k) = U ka (k+l) as comprising a sequence of three-term inner products. Analysing these in standard fashion we arrive at the equation |∆U k| < [(1 + u)4 - 1]|Uk|,
(21.20)
where we have taken into account the rounding errors in forming Since
(21.19) and (21.20) imply that (21.21)
Applying Lemma 3.7 to (21.21) and using (21.17), we obtain the desired bound for the forward error. The product |U0| . . . |Un-1||Ln-1| . . . |L0| in (21.18) is an upper bound for |U0. . .Un-1Ln-1 . . . L0| = |P–T| and is equal to it when there is no subtractive cancellation in the latter product. To gain insight, suppose the points are distinct and consider the case n = 3. We have
V ANDERMONDE S YSTEMS
436
(21.22)
There is no subtractive cancellation in this product as long as each matrix has the alternating (checkerboard) sign pattern defined, for A = (aij ), by (–l)i+jaij > 0. This sign pattern holds for the matrices Li if the points α i are arranged in increasing order. The matrices Ui have the required sign pattern provided that (in general) θi
>
0,
γi
>
0
for
all
i,
and
β i - αk < 0 for all i + k < n – 1.
In view of Table 21.2 we have the following result. Corollary 21.5. If 0 < α0 < α1 < · · · < α n then for the monomials, or the Chebyshev, Legendre, or Hermite polynomials,
Corollary 21.5 explains the high accuracy observed by Björck and Pereyra. Note that if |P - T ||f| < t n |P - T f | = t n |a| then, under the conditions of the corollary, which shows that the componentwise relative error is bounded by c(n, u)tn. For the problem of Björck and Pereyra it can be shown that tn n 4/24. Another factor contributing to the high accuracy in this problem is that many of the subtractions αj – αj-k-1 = 1/(j + 3) – 1/(j – k + 2) are performed exactly, in view of Theorem 2.5. Note that under the conditions of the corollary P-T has the alternating sign pattern, since each of its factors does. Thus if (–1)i fi > 0 then tn = 1, and the corollary implies that is accurate essentially to full machine precision, independent of the condition number In particular, taking f to be a column of the identity matrix shows that we can compute the inverse of P to high relative accuracy, independent of its condition number.
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21.3.2. Residual Next we look at the residual, r = f –
Rewriting (21.21),
(21.23) From the proof of Theorem 21.4 and the relation (5.9) it is easy to show that ( L k + ∆Lk)-l = L k-l + Ek,
|E k | < [(1 - u)-3 - 1]|Lk-1 |
Strictly, an analogous bound for (Uk + ∆Uk)-l does not hold, since ∆U k cannot be expressed in the form of a diagonal matrix times U k. However, it seems reasonable to make a simplifying assumption that such a bound is valid, say (U k + ∆Uk)-l = Uk-l + Fk,
|F k | < [(1 - u)-4 - 1]|Uk-1|.
(21.24)
Then, writing (21.23) as
and invoking Lemma 3.7, we obtain the following result. Theorem 21.6. Under the assumption (21.24), the residual of the computed solution from Algorithm 21.2 is bounded by (21.25) where d(n, u) := (1 – u) – 7n – 1 = 7nu + O(u2). For the monomials, with distinct, nonnegative points arranged in increasing order, the matrices Li and U i are bidiagonal with the alternating sign property, as we saw above. Thus Li- 1 > 0 and U i -1 > 0, and since PT = we obtain from (21.25) the following pleasing result, which guarantees a tiny componentwise relative backward error. Corollary 21.7. Let 0 < α0 < α1 < · · · < αn, and consider Algorithm 21.2 for the monomials. Under the assumption (21.24), the computed solution satisjies
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Table 21.3. Results for dual Chebyshev-Vandermonde-like system. n
10
20 -1
40
30 -7
2.5 x 10 2
6.3 X 10
6.0 X 10-13
1.1 x 10-7
4.7 X 10
-2
5.3 x 10-3
1.8 X 103 8.3 X 10-2
21.3.3. Dealing with Instability The potential instability of Algorithm 21.2 is illustrated by the following example. Take the Chebyshev polynomials Ti with the points αi = (the extrema of Tn ), and define the right-hand side by fi = (–1) i . The exact solution to PTa = f is the last column of the identity matrix. Relative errors -16 and residuals are shown in Table 21.3 (u = 10 ). Even though k2 (P) < 2 for all n (see Problem 21.7), the forward errors and relative residuals are large and grow with n. The reason for the instability is that there are large intermediate numbers in the algorithm (at the start of stage II for n = 40, is of order 1015); hence severe cancellation is necessary to produce the final answer of order 1. Looked at another way, the factorization of PT used by the algorithm is unstable because it is very sensitive to perturbations in the factors. How can we overcome this instability? There are two possibilites: prevention and cure. The only means at our disposal for preventing instability is to reorder the points αi . The ordering is arbitrary subject to condition (21.5) being satisfied. Recall that the algorithms construct an LU factorization of PT in factored form, and note that permuting the rows of PT is equivalent to reordering the points αi . A reasonable approach is therefore to take whatever ordering of the points would be produced by Gaussian elimination with T partial pivoting (GEPP) applied to P . Since the diagonal elements of U in T P = LU have the form
where ha depends only on the θi , and since partial pivoting maximizes the pivot at each stage, this ordering of the α i can be computed at the start of the algorithm in n2 flops (see Problem 21.8). This ordering is essentially the Leja ordering (5.13) (the difference is that partial pivoting leaves α0 unchanged). To attempt to cure observed instability we can use iterative refinement in fixed precision. Ordinarily, residual computation for linear equations is
21.3 S TABILITY
439
trivial, but in this context the coefficient matrix is not given explicitly and computing the residual turns out to be conceptually almost as difficult, and computationally as expensive, as solving the linear system! To compute the residual for the dual system we need a means for evaluating ψ(t) in (21.7) and its first k < n derivatives, where k = max i (i – r(i)) is the order of confluence. Since the polynomials pj satisfy a three-term recurrence relation we can use an extension of the Clenshaw recurrence formula (which evaluates ψ but not its derivatives). The following algorithm implements the appropriate recurrences, which are given by Smith [927, 1965] (see Problem 21.9). Algorithm 21.8 (extended Clenshaw recurrence). This algorithm computes the k + 1 values yj = ψ (j)(x), 0 < j < k, where ψ is given by (21.7) and k < n. It uses a work vector z of order k. Set y(0: k) = z(0: k) = 0 y0 = an for j = n – 1: – 1:0 temp = y0 y0 = θj (x – β j ) z0 = temp for i = 1: min(k, n – j) temp = yi yi = θj((x – β j)yi + zi-1) – γj + l z i zi = temp end end m=1 for i = 2:k m=m*i yi = m * yi end Computing the residual using Algorithm 21.8 costs between 3n2 flops (for full confluence) and 6n2 flops (for the nonconfluent case); recall that Algorithm 21.2 costs at most 9n 2 /2 flops! The use of iterative refinement can be justified with the aid of Theorem 11.3. For (confluent) Vandermonde matrices, the residuals are formed using Homer’s rule and (11 .7) holds in view of (5.3) and (5.7). Hence for standard Vandermonde matrices, Theorem 11.3 leads to an asymptotic componentwise backward stability result. A complete error analysis of Algorithm 21.8 is not available for (confluent) Vandermonde-like matrices, but it is easy to see that (11.7) will not always hold. Nevertheless it is clear that
440
V ANDERMONDE S YSTEMS
a normwise bound can be obtained (see Oliver [805, 1977] for the special case of the Chebyshev polynomials) and hence an asymptotic normwise stability result can be deduced from Theorem 11.3. Thus there is theoretical backing for the use of iterative refinement with Algorithm 21.8. Numerical experiments using Algorithm 21.8 in conjunction with the partial pivoting reordering and fixed precision iterative refinement show that both techniques are effective means for stabilizing the algorithms, but that iterative refinement is likely to fail once the instability y is sufficiently severe. Because of its lower cost, the reordering approach is preferable. Two heuristics are worth noting. Consider a (confluent) Vandermondelike system Px = b Heuristic 1: it is systems with a large normed solution that are solved to high accuracy by the fast algorithms. To produce a large solution the algorithms must sustain little cancellation, and the error analysis shows that cancellation is the main cause of instability. Heuristic 2: GEPP is unable to solve accurately Vandermonde systems with a very large normed solution The pivots for GEPP will tend to satisfy so that the computed solution will tend to satisfy A consequence of these two heuristics is that for Vandermonde(-like) systems with a very large-normed solution the fast algorithms will be much more accurate (but no more backward stable) than GEPP. However, we should be suspicious of any framework in which such systems arise; although the solution vector may be obtained accurately (barring overflow), subsequent computations with numbers of such a wide dynamic range will probably themselves be unstable.
21.4. Notes and References The formulae (21. 1) and (21.2), and inversion methods based on these formulae, have been discovered independently by many authors. Traub [1013, 1966, 14] gives a short historical survey, his earliest reference being a 1932 book by Kowalewski. There does not appear to be any published error analysis for Algorithm 21.1 (see Problem 21.3). There is little justification for using the output of the algorithm to solve the primal or dual linear system, as is done in [842, 1992, §2.8]; Algorithms 21.2 and 21.3 are more efficient and almost certainly at least as stable. Calvetti and Reichel [179, 1993] generalize Algorithm 21.1 to Vandermonde-like matrices, but they do not present any error analysis. Gohberg and Olshevsky [456, 1994] give another O(n2) flops algorithm for inverting a Chebyshev–Vandermonde matrix. The standard condition number K(V) is not an appropriate measure of sensitivity when only the points αi are perturbed, because it does not reflect the special structure of the perturbations. Appropriate condition numbers were first derived by Higham [533, 1987] and are comprehensively investigated
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by Bartels and D. J. Higham [76, 1992]; see Problem 21.10. Methods for solving the dual and primal Vandermonde systems have an interesting history. The earliest algorithm was derived by Lyness and Moler [718, 1966] via Neville interpolation; it solves the dual system in O(n3) flops. The first O(n2 ) algorithm was obtained by Ballester and Pereyra [52, 1967]; it computes the LU factors of the Vandermonde matrix and requires O(n2) elements of storage. Björck and Pereyra [121, 1970] derived the specialization of Algorithms 21.2 and 21.3 to nonconfluent Vandermonde matrices; these algorithms require no storage other than that for the problem data. The algorithms of Björck and Pereyra were generalized by Björck and Elfving to confluent systems [117, 1973], and by Higham to Vandermonde-like systems [536, 1988] and confluent Vandermonde-like systems [547, 1990]. The error analysis in this chapter is taken from [547, 1990]. Tang and Golub [992, 1981] give a block algorithm that requires only real arithmetic to solve a Vandermonde system in which all the points appear in complex conjugate pairs. Other O(n2 ) algorithms for solving Chebyshev-Vandermonde systems are given by Reichel and Opfer [865, 1991] and Calvetti and Reichel [178, 1992]. The former algorithms are progressive, in that they allow the solution to be updated when a new point αi is added; they generalize progressive algorithms of Björck and Pereyra [121, 1970]. Boros, Kailath, and Olshevsky [135, 1994] use the concept of displacement structure to derive further O(n2) algorithms for solving Vandermonde and Chebyshev–Vandermonde systems. No error analysis is given in [135, 1994], [178, 1992], or [865, 1991]. The O(n2) complexity of the algorithms mentioned above for solving Vandermonde-like systems is not optimal. Lu [714, 1994], [715, 1995], [716, 1996] derives O(n log n log p) flops algorithms, where p is the number of distinct points. The numerical stability and practical efficiency of the algorithms remain to be determined. Bini and Pan [98, 1994] give an O ( n log2 n) algorithm for solving a dual Vandermonde system that involves solving related Toeplitz and Hankel systems. Since Vandermonde systems can be solved in less than O(n3) flops it is natural to ask whether the O(mn 2 ) complexity of QR factorization of an m x n matrix can be bettered for a Vandermonde matrix. QR factorization algorithms with O(mn) flop counts have been derived by Demeure [277, 1989], [278, 1990], and for Vandermonde-like matrices where the polynomials satisfy a three-term recurrence by Reichel [864, 1991]. No error analysis has been published for these algorithms. Demeure’s algorithms are likely to be unstable, because they form the normal equations matrix VTV.
Problems 21.1. Derive a modified version of Algorithm 21.1 in which the scale factor
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s = q(αi ) is computed directly as
What is the flop count for this version? 21.2. (Calvetti and Reichel [179, 1993]) Generalize Algorithm 21.1 to the inversion of a Vandermonde-like matrix for polynomials that satisfy a threeterm recurrence relation. 21.3. Investigate the stability of Algorithm 21.1 and the modified version of Problem 21.1. (a) Evaluate the left and right residuals of the computed inverses; compare the results with those for GEPP. (b) Show that Algorithm 21.1 always performs subtractions of like-signed numbers and explain the implications for stability. (Does Algorithm 21.1 share the high accuracy properties discussed at the end of 521.3.1?) (c) (RESEARCH PROBLEM) Derive and explore forward error bounds and residual bounds for both algorithms. Extend the analysis to the algorithms of Calvetti and Reichel [179, 1993]. 21.4. By summing (21.1) for i = 0:n, show that What does this imply about the sign pattern of V–1 ? What is the sum of all the elements o f V –l ? be a Chebyshev–Vandermonde 21.5. Let T = T ( α0, α1, . . . , α n ) = matrix (Tj is the Chebyshev polynomial of degree j), with T–1 = (uij). Analogously to (21.1) we have Hence where V-l = V(α0, α1, . . . ,αn )-l = (wij). Show that
and hence that (Hint: show that T = LV for a lower triangular matrix L and use the fact that 21.6. Show that for a nonconfluent Vandermonde-like matrix P = where the pi satisfy (21.6),
(Hint: study the structure of (21.22 ).)
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21.7. Show that for the Chebyshev-Vandermonde-like matrix T = T(α 0 , α 1 , . . . , α n ), 1.
K 2 (T)
2.
K2 (T)
=
for αi = cos((i + ½)π/(n + 1)) (zeros of Tn+1 ).
< 2, for αi = cos(iπ/n ) (extrema of Tn ).
(Hint: use the discrete orthogonality properties of the Chebyshev polynomials; see, e.g., Hamming [501, 1973, pp.472–473].) 21.8. Derive an O(n2 ) flops algorithm that reorders the distinct points α0 , α1, . . . . αn according to the same permutation that would be produced by GEPP applied to PT(α0, α1,. . . . , αn ). (Cf. Problem 5.4. ) Can your algorithm ever produce the increasing ordering? 21.9. Derive Algorithm 21.8 by differentiating repeatedly the original Clenshaw recurrence (which is Algorithm 21.8 with k = 0) and resealing so as to consign factorial terms to a cleanup loop at the end. Derive an algorithm for computing the residual for the primal system in a similar way, using recurrences obtained by differentiating (21.6). 21.10. (Higham [533, 1987]) A structured condition number for the primal Vandermonde system Vx = b, where V = V(α0, α1, . . . . ,α n ), can be defined by
Show that cond(V) = and derive a corresponding condition number for the dual system VTa = f.
Previous Home
Chapter 22
Fast Matrix Multiplication
A simple but extremely valuable bit of equipment in matrix multiplication consists of two plain cards, with a re-entrant right angle cut out of one or both of them if symmetric matrices are to be multiplied. In getting the element of the ith row and jth column of the product, the ith row of the first factor and the jth column of the second should be marked by a card beside, above, or below it. — HAROLD HOTELLING, Some New Methods in Matrix Calculation (1943)
It was found that multiplication of matrices using punched card storage could be a highly efficient process on the Pilot ACE, due to the relative speeds of the Hollerith card reader used for input (one number per 16 ins.) and the automatic multiplier (2 ins.). While a few rows of one matrix were held in the machine the matrix to be multiplied by it was passed through the card reader. The actual computing and selection of numbers from store occupied most of the time between the passage of successive rows of the cards through the reader, so that the overall time was but little longer than it would have been if the machine had been able to accommodate both matrices. — MICHAEL WOODGER, The History and Present Use of Digital Computers at the National Physical Laboratory (1958)
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22.1. Methods A fast matrix multiplication method forms the product of two n x n matrices in arithmetic operations, where ω < 3. Such a method is more efficient asymptotically than direct use of the definition (22.1) which requires O(n3) operations. For over a century after the development of matrix algebra in the 1850s by Cayley, Sylvester and others, this definition provided the only known method for multiplying matrices. In 1967, however, to the surprise of many, Winograd found a way to exchange half the multiplications for additions in the basic formula [1105, 1968]. The method rests on the identity, for vectors of even dimension n, (22.2) When this identity is applied to a matrix product AB, with x a row of A and y a column of B, the second and third summations are found to be common to the other inner products involving that row or column, so they can be computed once and reused. Winograd’s paper generated immediate practical interest because on the computers of the 1960s floating point multiplication was typically two or three times slower than floating point addition. (On todays’ machines these two operations are usually similar in cost). Shortly after Winograd’s discovery, Strassen astounded the computer science community by finding an operations method for matrix multiplication (log2 7 2.807). A variant of this technique can be used to compute A-l (see Problem 22.8) and thereby to solve AX = b, both in operations. Hence the title of Strassen’s 1969 paper [962, 1969], which refers to the question of whether Gaussian elimination is asymptotically optimal for solving linear systems. Strassen’s method is based on a circuitous way to form the product of a pair of 2 x 2 matrices in 7 multiplications and 18 additions, instead of the usual 8 multiplications and 4 additions. As a means of multiplying 2 x 2 matrices the formulae have nothing to recommend them, but they are valid more generally for block 2 x 2 matrices. Let A and B be matrices of dimensions m x n and n x p respectively, where all the dimensions are even, and partition each of A, B, and C = AB into four equally sized blocks: (22.3)
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Strassen’s formulae are P1 = (A ll + A 22 )(B II + B 22 ) P2 = (A21 + A22 )B11 , P3 = A11 (B12 – B22 ), P4 P5 P6 P7
= = = =
A22 (B21 – B11 ), (A11 + A12 )B22 , (A 21 – A II )(B II + B 12 ) (A12 – A22 )(B21 + B22 )
(22.4)
C11 = P1 + P4 – P5 + P7, C12 = P3 + P5, C21 = P2 + P4, C22 = P1 + P3 – P2 + P6. Counting the additions (A) and multiplications (M) we find that while conventional multiplication requires mnpM + m(n – 1)pA, Strassen’s algorithm, using conventional multiplication at the block level, requires
Thus, if m, n, and p are large, Strassen’s algorithm reduces the arithmetic by a factor of about 7/8. The same idea can be used recursively on the multiplications associated with the Pa. In practice, recursion is only performed down to the “crossover” level at which any savings in floating point operations are outweighed by the overheads of a computer implementation. To state a complete operation count, we suppose that m = n = p = 2 k and that recursion is terminated when the matrices are of dimension no = 2 r , at which point conventional multiplication is used. The number of multiplications and additions can be shown to be M(k) = 7 k - r8r ,
(22.5)
The sum M(k) + A(k) is minimized over all integers r by r = 3; interestingly, this value is independent of k. The total operation count for the “optimal” no = 8 is less than
Hence, in addition to having a lower exponent, Strassen’s method has a reasonable constant.
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Winograd found a variant of Strassen’s formulae that requires the same number of multiplications but only 15 additions (instead of 18). This variant therefore has slightly smaller constants in the operation count for n x n matrices. For the product (22.3) the formulae are S1 = A21 + A22,
M1 = S 2 S 6 ,
S2 = S1 – A11,
M 2 = A 1l B 1l , T 2 = T 1 + M4,
S3 = A11 – A21, S4 = A12 – S2,
M 3 = A 12 B 21 , M4 = S 3 S 7 ,
S5 = B12 – B11,
M5 = S 1 S 5 ,
S6 = B22 – S5, S7 = B22 – B12,
M 6 = S 4 B 22 ,
C11 = M2 + M3, C12 = TI + M5 + M6,
M7 = A22 S 8 ,
C21 = T2 – M7,
S8 = S6 – B21,
T 1 = M1 + M2 ,
(22.6)
C22 = T2 + M5.
Until the late 1980s there was a widespread view that Strassen’s method was of theoretical interest only, because of its supposed large overheads for dimensions of practical interest (see, for example, [909, 1988]), and this view is still expressed by some [842, 1992]. However, in 1970 Brent implemented Strassen’s algorithm in Algol-W on an IBM 360/67 and concluded that in this environment, and with just one level of recursion, the method runs faster than the conventional method for n > 110 [142, 1970]. In 1988, Bailey compared his Fortran implementation of Strassen’s algorithm for the Cray-2 with the Cray library routine for matrix multiplication and observed speedup factors ranging from 1.45 for n = 128 to 2.01 for n = 2048 (although 35% of these speedups were due to Cray-specific techniques) [43, 1988]. These empirical results, together with more recent experience of various researchers, show that Strassen’s algorithm is of practical interest, even for n in the hundreds. Indeed, Fortran codes for (Winograd’s variant of) Straasen’s method have been supplied with IBM’s ESSL library [595, 1988] and Cray’s UNICOS library [602, 1989] since the late 1980s. Strassen’s paper raised the question “what is the minimum exponent ω such that multiplication of n x n matrices can be done in operations?” Clearly, ω > 2, since each element of each matrix must partake in at least one operation. It was 10 years before the exponent was reduced below Strassen’s log2 7 . A flurry of publications, beginning in 1978 with Pan and his exponent 2.795 [815, 1978], resulted in reduction of the exponent to the current record 2.376, obtained by Coppersmith and Winograd in 1987 [245, 1987]. Figure 22.1 plots exponent versus time of publication (not all publications are represented in the graph); in principle, the graph should extend back to 1850! Some of the fast multiplication methods are based on a generalization of Strassen’s idea to bilinear forms. Let A, B A bilinear noncommuta-
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Figure 22.1. Exponent versus time for matrix multiplication. tive algorithm over for multiplying A and B with t “nonscalar multiplications” forms the product C = AB according to (22.7a)
(22.7b) where the elements of the matrices W, U (k) , and V (k) are constants. This algorithm can be used to multiply n x n matrices A and B, where n = hk, as follows: partition A, B, and C into h 2 blocks Aij , Bij , and Cij of size h k–1 , then compute C = AB by the bilinear algorithm, with the scalars aij, bij, and cij replaced by the corresponding matrix blocks. (The algorithm is applicable to matrices since, by assumption, the underlying formulae do not depend on commutativity.) To form the t products Pk of (n/h) x (n/h) matrices, partition them into h 2 blocks of dimension n/h2 and apply the algorithm recursively. The total number of scalar multiplications required for the multiplication is tk = n α, where α = logh t. Strassen’s method has h = 2 and t =7. For 3 x 3 multiplication (h = 3), the smallest t obtained so far is 23 [683, 1976]; since log323 2.854 > log27, this does not yield any improvement over Strassen’s method. The method
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described in Pan’s 1978 paper has h = 70 and t = 143,640, which yields α = log70143,640 = 2.795 . . . . In the methods that achieve exponents lower than 2.775, various intricate techniques are used. Laderman, Pan, and Sha [684, 1992] explain that for these methods “very large overhead constants are hidden in the ’O’ notation’’, and that the methods “improve on Strassen’s (and even the classical) algorithm only for immense numbers N“. A further method that is appropriate to discuss in the context of fast multiplication methods, even though it does not reduce the exponent, is a method for efficient multiplication of complex matrices. The clever formula (a + ib)(c + id) = ac - bd + i[(a + b)(c + d) - ac - bd]
(22.8)
computes the product of two complex numbers using three real multiplications instead of the usual four. Since the formula does not rely on commutativity it extends to matrices. Let A = A1 + iA2 and B = Bl + iB2, where Aj, Bj and define C = C1 + iC2 = AB. Then C can be formed using three real matrix multiplications as T1 = A1 B1 ,
T2 = A2 B2 ,
C1 = T1 – T2,
(22.9)
C2 = (A1 + A2)(B1 + B2) – T1 – T2, which we will refer to as the "3M method”. This computation involves 3n 3 scalar multiplications and 3n3 + 2n2 scalar additions. Straightforward evaluation of the conventional formula C = A1B1 – A2B2 + i(A1B2 + A2B1) requires 4n3 multiplications and 4n3 – 2n2 additions. Thus, the 3M method requires strictly fewer arithmetic operations than the conventional means of multiplying complex matrices for n > 3, and it achieves a saving of about 25% for n > 30 (say). Similar savings occur in the important special case where A or B is triangular. This kind of clear-cut computational saving is rare in matrix computations! IBM’s ESSL library and Cray’s UNICOS library both contain routines for complex matrix multiplication that apply the 3M method and use Strassen’s method to evaluate the resulting three real matrix products.
22.2. Error Analysis To be of practical use, a fast matrix multiplication method needs to be faster than conventional multiplication for reasonable dimensions without sacrificing numerical stability. The stability properties of a fast matrix multiplication method are much harder to predict than its practical efficiency, and need careful investigation.
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The forward error bound (3.12) for conventional computation of C = AB, where A, B can be written (22.10 Miller [756, 1975] shows that any polynomial algorithm for multiplying n x n matrices that satisfies a bound of the form (22.10) (perhaps with a different constant) must perform at least n 3 multiplications. (A polynomial algorithm is essentially one that uses only scalar addition, subtraction, and multiplication.) Hence Strassen’s method, and all other polynomial algorithms with an exponent less than 3, cannot satisfy (22.10). Miller also states, without proof, that any polynomial algorithm in which the multiplications are all of the form must satisfy a bound of the form (22.11) It follows that any algorithm based on recursive application of a bilinear noncommutative algorithm satisfies (22.11); however, the all-important constant f n is not specified. These general results are useful because they show us what types of results we can and cannot prove and thereby help to focus our efforts. In the subsections below we analyse particular methods. Throughout the rest of this chapter an unsubscripted matrix norm denotes
As noted in 6.2, this is not a consistent matrix norm, but we do have the bound ||AB|| < n||A|| ||B|| for n x n matrices. 22.2.1. Winograd’s Method Winograd’s method does not satisfy the conditions required for the bound (22.11), since it involves multiplications with operands of the form aij + brs. However, it is straightforward to derive an error bound. Theorem 22.1 (Brent). Let x, y where n is even. The inner product computed by Winograd’s method satisfies (22.12) Proof. A straightforward adaptation of the inner product error analysis in 3.1 produces the following analogue of (3.3):
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where the
and β i are all bounded in modulus by γn /2+4.
Hence
The analogue of (22.12) for matrix multiplication is ||AB – fl(AB)|| < Conventional evaluation of xTy yields the bound (see (3.5)) (22.13) The bound (22. 12) for Winograd’s method exceeds the bound (22.13) by a factor approximately Therefore Winograd’s method is stable if have similar magnitude, but potentially unstable if they differ widely in magnitude. The underlying reason for the instability is that Winograd’s method relies on cancellation of terms x2i–1 x2i and y2i–ly2i that can be much larger than the final answer therefore the intermediate rounding errors can swamp the desired inner product. A simple way to avoid the instability is to scale x µ x and y µ-ly before applying Winograd’s method, where µ, which in practice might be a power of the machine base to avoid roundoff, is chosen so that When using Winograd’s method for a matrix multiplication AB it suffices to carry out a single scaling A µA and B µ -lB such that ||A|| ||B||. If −1 A and B are scaled so that τ < ||A||/||B|| < τ then
22.2.2. Strassen’s Method Until recently there was a widespread belief that Strassen’s method is numerically unstable. The next theorem, originally proved by Brent in 1970, shows that this belief is unfounded.
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Theorem 22.2 (Brent). Let A, B where n = 2 k. Suppose that C = AB is computed by Strassen’s method and that n0 = 2 r is the threshold at which conventional multiplication is used. The computed product satisfies (22.14) Proof. We will use without comment the norm inequality ||AB|| < n||A|| ||B|| = 2 k||A|| ||B||. Assume that the computed product AB from Strassen’s method satisfies (22.15) ||E|| < cku||A|| ||B|| + O(u2), = AB + E, where ck is a constant. In view of (22.10), (22.15) certainly holds for n = no, with c r = Our aim is to verify (22.15) inductively and at the same time to derive a recurrence for the unknown constant ck. Consider Cll in (22.4), and, in particular, its subterm P1. Accounting for the errors in matrix addition and invoking (22.15), we obtain
where | ∆A| < u|A11 + A22|, |∆B | < u|B11 + B22 |, ||E1|| < ck-1u||A11 + A22 + ∆A|| ||Bll + B22 + ∆B|| + O(u2) < 4ck-1u||A|| ||B|| + O(u2). Hence = P1 + F1, ||F1|| < (8 · 2 k-1 + 4c k - 1)u||A|| ||B|| + O(u2). Similarly, where
= A 22 (B 21 – B ll + ∆B) + E4, |∆B| < u|B21 – B11|, ||E4 || < ck-1u||A22 || ||B21 - B11 + ∆B|| + O(u2),
which gives = P4 + F4, ||F4|| < (2 · 2 k-1 + 2c k - 1) u||A|| ||B|| + O(u2). Now
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where =: P5 + F5 and =: P7 + F7 satisfy exactly the same error bounds as and respectively. Assuming that these four matrices are added in the order indicated, we have
Clearly, the same bound holds for the other three ||∆Cij|| terms. Thus, overall, = AB + ∆C ,
||∆C|| < (46 · 2 k-1 + 12ck- l )u||A|| ||B|| + O(u2).
A comparison with (22.15) shows that we need to define the ck by c k = 12ck-1 +46 · 2 k-1, k > r, cr = 4 r , where cr =
(22.16)
Solving this recurrence we obtain
which gives (22.14). The forward error bound for Strassen’s method is not of the componentwise form (22.10) that holds for conventional multiplication, which we know it cannot be by Miller’s result. One unfortunate consequence is that while the scaling AB (AD)(D-1 B), where D is diagonal, leaves (22.10) unchanged, it can alter (22.14) by an arbitrary amount. The reason for the scale dependence is that Strassen’s method adds together elements of A matrix-wide (and similarly for B); for example, in (22.4) A11 is added to A22 , A12 , and A21 . This intermingling of elements is particularly undesirable when A or B has elements of widely differing magnitudes because it allows large errors to contaminate small components of the product. This phenomenon is well illustrated by the example
which is evaluated exactly in floating point arithmetic if we use conventional multiplication. However, Strassen’s method computes
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Because c22 involves subterms of order unity, the error c22 – will be of order u. Thus the relative error |c22 – |/|c22| = which is much larger than u if ε is small. This is an example where Strassen’s method does not satisfy the bound (22.10). For another example, consider the product X = P32 E, where Pn is the n x n Pascal matrix (see 26.4) and eij = 1/3. With just one level of recursion in Strassen’s method we find in MATLAB that is of order 10-5, so that, again, some elements of the computed product have high relative error. It is instructive to compare the bound (22.14) for Strassen’s method with the weakened, normwise version of (22.10) for conventional multiplication: (22.17) The bounds (22. 14) and (22. 17) differ only in the constant term. For Strassen’s method, the greater the depth of recursion the bigger the constant in (22.14): if we use just one level of recursion (n0 = n/2) then the constant is 3n2 + 25n, whereas with full recursion (n0 = 1) the constant is 6n3.585 – 5n. It is also interesting to note that the bound for Strassen’s method (minimal for n 0 = n) is not correlated with the operation count (minimal for n0 = 8). Our conclusion is that Strassen’s method has less favorable stability properties than conventional multiplication in two respects: it satisfies a weaker error bound (normwise rather than componentwise) and it has a larger constant in the bound (how much larger depending on no). Another interesting property of Strassen’s method is that it always involves some genuine subtractions (assuming that all additions are of nonzero terms). This is easily deduced from the formulae (22.4). This makes Strassen’s method unattractive in applications where all the elements of A and B are nonnegative (for example, in Markov processes). Here, conventional multiplication yields low componentwise relative error because, in (22.10), |A||B| = |AB| = |C|, yet comparable accuracy cannot be guaranteed for Strassen’s method. An analogue of Theorem 22.2 holds for Winograd’s variant of Strassen’s method. Theorem 22.3. Let A, B where n = 2 k. Suppose that C = AB is computed by Winograd’s variant (22.6) of Strassen’s method and that n0 = 2 r is the threshold at which conventional multiplication is used. The computed product satisfies (22.18) Proof. The proof is analogous to that of Theorem 22.2, but more tedious. It suffices to analyse the computation of C12, and the recurrence corresponding
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456 to (22.16) is c
k
= 1 8c k - l + 8 9 · 2 k – 1 , k > r, c r = 4 r .
Note that the bound for the Winograd–Strassen method has exponent log2 18 4.170 in place of log2 12 3.585 for Strassen’s method, suggesting that the price to be paid for a reduction in the number of additions is an increased rate of error growth. All the comments above about Strassen’s method apply to the Winograd variant. Two further questions are suggested by the error analysis: l l
How do the actual errors compare with the bounds? Which formulae are the more accurate in practice, Strassen’s or Winograd’s variant?
To give some insight we quote results obtained with a single precision Fortran 90 implementation of the two methods (the code is easy to write if we exploit the language’s dynamic arrays and recursive procedures). We take random n x n matrices A and B and look at ||AB – fl(AB)||/(||A|| ||B||) for n0 = l, 2, . . . , 2k = n (note that this is not the relative error, since the denominator is ||A|| ||B|| instead of ||AB||, and note that no = n corresponds to conventional multiplication). Figure 22.2 plots the results for one random matrix of order 1024 from the uniform [0, 1] distribution and another matrix of the same size from the uniform [– 1, 1] distribution. The error bound (22.14) for Strassen’s method is also plotted. Two observations can be made. l
Winograd’s variant can be more accurate than Strassen’s formulae, for all no, despite its larger error bound.
l
The error bound overestimates the actual error by a factor up to 1.8 x 106 for n = 1024, but the variation of the errors with no is roughly as predicted by the bound.
22.2.3. Bilinear Noncommutative Algorithms Bini and Lotti [97, 1980] have analysed the stability of bilinear noncommutative algorithms in general. They prove the following result. Theorem 22.4 (Bini and Lotti). Let A, B (n = h k ) and let the product C = AB be formed by a recursive application of the bilinear noncommutative algorithm (22.7), which multiplies h x h matrices using t nonscalar multiplications. The computed product satisfies (22.19)
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Figure 22.2. Errors for Strassen’s method with two random matrices of dimension n = 1024. Strassen’s formulae: “x”, Winograd’s variant: "o". X-axis contains log2 of recursion threshold n0, 1 < n0 < n. Dot-dash line is error bound for Strassen’s formulae.
where α and β are constants that depend on the number of nonzero terms in the matrices U, V and W that define the algorithm. The precise definition of α and β is given in [97, 1980]. If we take k = 1, so that h = n, and if the basic algorithm (22.7) is chosen to be conventional multiplication, then it turns out that α = n – 1 and β = n, so the bound of the theorem becomes (n – 1)nu||A|| ||B|| + O(u2), which is essentially the same as (22.17). For Strassen’s method, h = 2 and t = 7, and α = 5, β = 12, which so the theorem produces the bound is a factor log2 n larger than (22.14) (with n0 = 1). This extra weakness of the bound is not surprising given the generality of Theorem 22.4. Bini and Lotti consider the set of all bilinear noncommutative algorithms that form 2 x 2 products in 7 multiplications and that employ integer constants of the form ±2i , where i is an integer (this set breaks into 26 equivalence classes). They show that Strassen’s method has the minimum exponent β in its error bound in this class (namely, β = 12). In particular, Winograd's variant of Strassen’s method has β = 18, so Bini and Lotti’s bound has the same exponent log2 18 as in Theorem 22.3.
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458 22.2.4. The 3M Method
A simple example reveals a fundamental weakness of the 3M method. Consider the computation of the scalar
In floating point arithmetic, if y is computed in the usual way, as y = θ(1/θ ) + ( 1 /θ) θ, then no cancellation occurs and the computed has high relative The 3M method computes accuracy:
If |θ| is large this formula expresses a number of order 1 as the difference will almost certainly be contaminated of large numbers. The computed by rounding errors of order uθ2, in which case the relative error is large: However, if We measure the error in relative to z, then it is acceptably small: This example suggests that the 3M method may be stable, but in a weaker sense than for conventional multiplication. To analyse the general case, consider the product C 1 + iC 2 = (A1 + iA2 )( B1 + iB2 ), where Ak, Bk, Ck k = 1:2. Using (22.10) we find that the computed product from conventional multiplication,
satisfies (22.20) (22.21) For the 3M method C1 is computed in the conventional way, and so (22.20) holds. It is straightforward to show that satisfies (22.22) Two notable features of the bound (22.22) are as follows. First, it is of a different and weaker form than (22.21); in fact, it exceeds the sum of the bounds (22.20) and (22.21). Second and more pleasing, it retains the property of (22.20) and (22.21) of being invariant under diagonal scalings C = AB
D 1 AD 2 · D 2 -1 BD 3 = D 1 CD 3 ,
D j diagonal,
in the sense that the upper bound ∆C2 in (22.22) scales also according to D 1 ∆C2D3. (The “hidden” second-order terms in (22.20)–(22.22) are invariant under these diagonal scalings. )
22.3 N OTES
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The disparity between (22.21) and (22.22) is, in part, a consequence of the differing numerical cancellation properties of the two methods. It is easy to show that there are always subtractions of like-signed numbers in the 3M method, whereas if A1, A2, Bl, and B2 have nonnegative elements (for example) then no numerical cancellation takes place in conventional multiplication. We can define a measure of stability with respect to which the 3M method matches conventional multiplication by taking norms in (22.21) and (22.22). We obtain the weaker bounds (22.23) (22.24) (having used || |A1| + |A2| || < ||Al + iA2 ||). Combining these with an analogous weakening of (22.20), we find that for both conventional multiplication and the 3M method the computed complex matrix satisfies
where cn = O(n). The conclusion is that the 3M method produces a computed product whose imaginary part may be contaminated by relative errors much larger than those for conventional multiplication (or, equivalently, much larger than can be accounted for by small componentwise perturbations in the data A and B). However, if the errors are measured relative to ||A|| ||B||, which is a natural quantity to use for comparison when employing matrix norms, then they are just as small as for conventional multiplication. It is straightforward to show that if the 3M method is implemented using Strassen’s method to form the real matrix products, then the computed complex product satisfies the same bound (22.14) as for Strassen’s method itself, but with an extra constant multiplier of 6 and with 4 added to the expression inside the square brackets.
22.3. Notes and References A good introduction to the construction of fast matrix multiplication methods is provided by the papers of Pan [816, 1984] and Laderman, Pan, and Sha [684, 1992]. Harter [504, 1972] shows that Winograd’s formula (22.2) is the best of its kind, in a sense made precise in [504, 1972]. How does one derive formulae such as those of Winograd and Strassen, or that in the 3M method? Inspiration and ingenuity seem to be the key. A fairly straightforward, but still not obvious, derivation of Strassen’s method is given by Yuval [1124, 1978]. Gustafson and Aluru [491, 1996] develop algorithms
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that systematically search for fast algorithms, taking advantage of a parallel computer. In an exhaustive search taking 21 hours of computation time on a 256 processor nCUBE 2, they were able to find 12 methods for multiplying 2 complex numbers in 3 multiplications and 5 additions; they could not find a method with fewer additions, thus proving that such a method does not exist. However, they estimate that a search for Strassen’s method on a 1024 processor nCUBE 2 would take many centuries, even using aggressive pruning rules, so human ingenuity is not yet redundant! To obtain a useful implementation of Strassen’s method a number of issues have to be addressed, including how to program the recursion, how best to handle rectangular matrices of arbitrary dimension (since the basic method is defined only for square matrices of dimension a power of 2), and how to cater for the extra storage required by the method. These issues are discussed by Bailey [43, 1988], Bailey, Lee, and Simon [47, 1991], Fischer [374, 1974], Higham [544, 1990], Kreczmar [673, 1976], and [934, 1976], among others. Douglas, Heroux, Slishman, and Smith [317, 1994] present a portable Fortran implementation of Winograd’s variant of Strassen’s method for real and complex matrices, with a level-3 BLAS interface; they take care to use a minimal amount of extra storage (about 2n 3/3 elements of extra storage is required when multiplying n x n matrices). Higham [544, 1990] shows how Strassen’s method can be used to produce algorithms for all the level-3 BLAS operations that are asymptotically faster than the conventional algorithms. Most of the standard algorithms in numerical linear algebra remain stable (in an appropriately weakened sense) when fast level-3 BLAS are used. See, for example, Chapter 12, $18.4, and Problems 11.4 and 13.2. Knight [664, 1995] shows how to choose the recursion threshold to minimize the operation count of Strassen’s method for rectangular matrices. He also shows how to use Strassen’s method to compute the QR factorization of an m x n matrix in O(mn1.838 ) operations instead of the usual O(mn2 ). Bailey, Lee, and Simon [47, 1991] substituted their Strassen’s method code for a conventionally coded BLAS3 subroutine SGEMM and tested LAPACK’S LU factorization subroutine SGETRF on a Cray Y-MP. They obtained speed improvements for matrix dimensions 1024 and larger. The Fortran 90 standard includes an intrinsic function MATMUL that returns the product of its two matrix arguments. The standard does not specify which method is to be used for the multiplication. An IBM compiler supports the use of Winograd’s variant of Strassen’s method, via an optional third argument to MATMUL (an extension to Fortran 90) [318, 1994], Brent was the first to point out the possible instability of Winograd’s method [143, 1970]. He presented a full error analysis (including Theorem 22. 1) and showed that scaling ensures stability. An error analysis of Strassen’s method was given by Brent in 1970 in
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an unpublished technical report that has not been widely cited [142, 1970]. Section 22.2.2 is based on Higham [544, 1990]. According to Knuth, the 3M formula was suggested by P. Ungar in 1963 [668, 1981, p. 647]. It is analogous to a formula of Karatsuba and Ofman [643, 1963] for squaring a 2n-digit number using three squarings of n-digit numbers. That three multiplications (or divisions) are necessary for evaluating the product of two complex numbers was proved by Winograd [1106, 1971]. Section 22.2.4 is based on Higham [552, 1992]. The answer to the question “What method should we use to multiply complex matrices?” depends on the desired accuracy and speed. In a Fortran environment an important factor affecting the speed is the relative efficiency of real and complex arithmetic, which depends on the compiler and the computer (complex arithmetic is automatically converted by the compiler into real arithmetic). For a discussion and some statistics see [552, 1992]. The efficiency of Winograd’s method is very machine dependent. Bjørstad, Marine, Sørevik, and Vajteršic [122, 1992] found the method useful on the MasPar MP-1 parallel machine, on which floating point multiplication takes about three times as long as floating point addition at 64-bit precision. They also implemented Strassen’s method on the MP-1 (using Winograd’s method at the bottom level of recursion) and obtained significant speedups over conventional multiplication for dimensions as small as 256. As noted in 22.1, Strassen [962, 1969] gave not only a method for multiplying n x n matrices in operations, but also a method for inverting an n x n matrix with the same asymptotic cost. The method is described in Problem 22.8. For more on Strassen’s inversion method see $24.3.2, Bailey and Ferguson [41, 1988], and Bane, Hansen, and Higham [51, 1993].
Problems 22.1. (Knight [664, 1995]) Suppose we have a method for multiplying n x n matrices in operations, where 2 < α < 3. Show that if A is m x n and operations, B is n x p then the product AB can be formed in where nl = min(m, n, p) and n2 and n3 are the other two dimensions. 22.2. Work out the operation count for Winograd’s method applied to n x n matrices. 22.3. Let S n (n0 ) denote the operation count (additions plus multiplications) for Strassen’s method applied to n x n matrices, with recursion down to the level of no x no matrices. Assume that n and no are powers of 2. For large n, estimate Sn (8)/ Sn(n) and Sn(1)/S n (8) and explain the significance of these ratios (use (22.5)). 22.4. (Knight [664, 1995]) Suppose that Strassen’s method is used to multiply
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an m x n matrix by an n x p matrix, where m = a2 j, n = b2 j, p = c2 j, and that conventional multiplication is used once any dimension is 2 r or less. Show that the operation count is α7 j + β4 j, where
Show that by setting r = 0 and a = 1 a special case of the result of Problem 22.1 is obtained. 22.5. Compare and contrast Winograd’s inner product formula for n = 2 with the imaginary part of the 3M formula (22.8). 22.6. Prove the error bound described at the end of tion of the 3M method and Strassen’s method.
22.2.4 for the combina-
22.7. Two fast ways to multiply complex matrices are (a) to apply the method to the original matrices and to use Strassen’s method to form three real matrix products, and (b) to use Strassen’s method with the method applied at the bottom level of recursion. Investigate the merits of two approaches with respect to speed and storage.
3M the 3M the
22.8. Strassen [962, 1969] gives a method for inverting an n x n matrix in operations. Assume that n is even and write
The inversion method is based on the following formulae:
The matrix multiplications are done by Strassen’s method and the inversions determining P1 and P6 are done by recursive invocations of the method itself. (a) Verify these formulae, using a block LU factorization of A, and show that they permit the claimed complexity. (b) Show that if A is upper triangular, Strassen’s method is equivalent to (the unstable) Method 2B of 13.2.2. (For a numerical investigation into the stability of Strassen’s inversion method, see 24.3.2.)
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22.9. Find the inverse of the block upper triangular matrix
Deduce that matrix multiplication can be reduced to matrix inversion. 22.10. (RESEARCH PROBLEM) Carry out extensive numerical experiments to test the accuracy of Strassen’s method and Winograd’s variant (cf. the results at the end of 22.2.2).
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Chapter 23
The Fast Fourier Transform and Applications
Once the [FFT] method was established it became clear that it had a long and interesting prehistory going back as far as Gauss. But until the advent of computing machines it was a solution looking for a problem. Fourier Analysis (1988)
Life as we know it would be very different without the FFT. — CHARLES F. VAN LOAN, Computation/ Frameworks for the Fast Fourier Transform (1992)
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23.1. The Fast Fourier Transform The matrix–vector product y = Fn x, where
is the key computation in the numerical evaluation of Fourier transforms. If the product is formed in the obvious way then O(n2) operations are required. The fast Fourier transform (FFT) is a way to compute y in just O ( n log n) operations. This represents a dramatic reduction in complexity. The FFT is best understood (at least by a numerical analyst!) by interpreting it as the application of a clever factorization of the discrete Fourier transform (DFT) matrix Fn . Theorem 23.1 (Cooley–Tukey radix 2 factorization). If n = 2 t then the DFT matrix Fn may be factorized as Fn = At . . . A1Pn,
(23.1)
where Pn is a permutation matrix and
Proof. See Van Loan [1044, 1992, Thm. 1.3.3]. The theorem shows that we can write y = Fnx as y = At . . . A1Pnx. which is formed as a sequence of matrix-vector products. It is the sparsity of the Ak (two nonzeros per row) that yields the O ( n log n) operation count. We will not consider the implementation of the FFT, and therefore we do not need to define the “bit reversing: permutation matrix Pn in (23.1). However, the way in which the weights are computed does affect the accuracy. We will assume that computed weights are used that satisfy, for all j and k, (23.2) Among the many methods for computing the weights are ones for which we can take µ = cu, µ = cu log j, and µ = cuj, where c is a constant that depends on the method; see Van Loan [1044, 1992, 1.4]. We are now ready to prove an error bound.
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Theorem 23.2. Let = fl(F n x), where n = 2t, be computed using the Cooley-Tukey radix 2 FFT, and assume that (23.2) holds. Then
Proof. Denote by Then
the matrix defined in terms of the computed
using the fact that each Ak has only two nonzeros per row, and recalling that we are using complex arithmetic. In view of (23.2),
Hence, overall,
Invoking Lemma 3.7 we find that
using Lemma 3.1 for the second inequality. Now Hence (23.3) Finally, because Fn is
times a unitary matrix
Theorem 23.2 says that the Cooley–Tukey radix 2 FFT yields a tiny forward error, provided that the weights are computed stably. It follows immediately that the computation is backward stable, since = y + ∆y = Fnx + ∆y implies = F n ( x + ∆x) with ||∆x||2/||x||2 = ||∆y||2/||y||2. If we form y = Fn x by conventional multiplication using the exact Fn , then (ProbHence when lem 3.7) so µ is of order u, the FFT has an error bound smaller than that for conventional multiplication by the same factor as the reduction in complexity of the method. To sum up, the FFT is perfectly stable.
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Figure 23.1. Error in FFT followed by inverse FFT (“o”). Dotted line is error bound. The inverse transform x = F n - 1 y = n – 1 F n *y can again be computed in O(n log n) operations using the Cooley–Tukey radix 2 factorization, and satisfies the same bound as in Theorem 23.2. (Strictly, we should replace t by t + 1 in the bound to account for the rounding error in dividing by n) For other variations of the FFT, based on different radices or different factorization of Fn , results analogous to Theorem 23.2 hold. A simple way to test the error bounds is to compute the FFT followed by the inverse transform, and to evaluate e n = ||x – Our analysis gives the bound en < 2n1/2 log2 nη + O(η 2). Figure 23.1 plots en and the approximate error bound n1/2 log2 nu for n = 2k , k = 0:16, with random x from the normal N(0,1) distribution (the FFTs were computed using MATLAB’s fft and ifft functions). The errors grow at roughly the same rate as the bound and are on average about a factor of 10 smaller than the bound.
23.2. Circulant Linear Systems A circulant matrix (or circulant, for short) is a special Toeplitz matrix in which the diagonals wrap around:
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Circulant matrices have the important property that they are diagonalized by the DFT matrix Fn : Fn CFn -l = D = diag(d i). Moreover, the eigenvalues are given by d = Fnc. Hence a linear system Cx = b can be solved in O(n log n) operations with the aid of the FFT as follows: (1) (2) (3) (4)
d = Fnc, g = Fnb, h = D -l g, x = Fn-1h.
The computation involves two FFTs, a diagonal scaling, and an inverse FFT. We now analyse the effect of rounding errors. It is convenient to write the result of Theorem 23.2 in the equivalent form (from (23.3)) (23.4) Steps (1) and (2) yield = (Fn + ∆1 )c, = (Fn + ∆2 )b,
||∆1||2 < f(n, u), ||∆2||2 < f(n, u).
(23.5)
Equation (23.5) implies that = D + ∆ 0 , ||∆ 0 || 2 < f(n, u)||C||2. For steps (3) and (4) we have, using Lemma 3.5,
Putting these equations together we have (23.6) or, working to first order,
We obtain a backward stability result.
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Theorem 23.3. Let C be a circulant and suppose the system Cx = b is solved by the FFT process described above, where the FFT satisfies (23.4). Then the computed solution satisfies (C + ∆C ) = b, where
The conclusion is that the FFT method for solving a circulant system is normwise backward stable provided that the FFT itself is computed stably. Although we have shown that solves a slightly perturbed system, the perturbed matrix C + ∆C is not a circulant. In fact, does not, in general, solve a nearby circulant system, as can be verified experimentally by computing the "(circulant backward error” using techniques from [527, 1992]. The basic reason for the lack of this stronger form of stability is that there are too few parameters in the matrix onto which to throw the backward error. A forward error bound can be obtained by applying standard perturbation theory to Theorem 23.3: the forward error is bounded by a multiple of k2 (C)u. That the forward error can be as large as k2 (C)u is clear from the analysis above, because (23.5) shows that the computed eigenvalue is contaminated by an error of order u
23.3. Notes and References For a unified treatment of the many different versions of the FFT, including implementation details, see Van Loan [1044, 1992]. For a comprehensive survey of the discrete Fourier transform see Briggs and Henson [148, 1995]. The Cooley–Tukey radix 2 FFT was presented in [240, 1965], which is one of the most cited mathematics papers of all time [554, 1993, p. 171]. The history of the FFT is discussed by Cooley [238, 1990], [239, 1994] and Cooley and Tukey [241, 1993]. Cooley [238, 1990] states that the earliest known reference to the FFT is an obscure 1866 paper of Gauss in neoclassic Latin, and he recommends that researchers not publish papers in neoclassic Latin! Theorem 23.2 is not new, but the proof is more concise than most in the literature. Early references that derive error bounds using the matrix factorization formulation of the FFT are Gentleman and Sande [434, 1966] and Ramos [859, 1971]. A full list of references for error analysis of the FFT is given by Van Loan [1044, 1992, 1.4]. Linzer [708, 1992] shows that the FFT-based circulant solver is forward stable and poses the question of whether or not the solver is backward stable. Our backward error analysis answers this question positively and therefore also proves forward stability.
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One application of circulant linear systems is in the preconditioned conjugate gradient method for solving Toeplitz systems. The idea of using a circulant preconditioned was suggested by Strang [960, 1986], and the theory and practice of this technique is now well developed. For more details see Chan, Nagy, and Plemmons [191, 1994] and the references therein. A good source of results about circulant matrices is the book by Davis [266, 1979].
Problems 23.1. (Bailey [44, 1993]) In high-precision multiplication we have two integer n -vectors x and y representing high-precision numbers and we wish to form the terms By padding the vectors with n zeros, these products can be expressed in the form where k + 1 – j is interpreted as k + 1 – j + n if k + 1 – j is negative. These products represent a convolution: a matrix–vector product involving a circulant matrix. Analogously to the linear system solution in 23.2, this product can be evaluated in terms of discrete Fourier transforms as z = where the dot denotes componentwise multiplication of two vectors. Since x and y are integer vectors, the zi should also be integers, but in practice they will be contaminated by rounding errors. Obtain a bound on z – and deduce a sufficient condition for the “nearest integer vector to to be the exact z.
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Chapter 24
Automatic Error Analysis
Given the pace of technology, I propose we leave math to the machines and go play outside. — CALVIN, Calvin and Hobbes by Bill Watterson (1992)
To analyse a given numerical algorithm we proceed as follows. A number which measures the effect of roundoff error is assigned to each set of data. “Hill-climbing” procedures are then applied to search for values large enough to signal instability. — WEBB MILLER, Software for Roundoff Analysis (1975)
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Automatic error analysis is any process in which we use the computer to help us analyse the accuracy or stability of a numerical computation. The idea of automatic error analysis goes back to the dawn of scientific computing. For example, running error analysis, described in 3.3, is a form of automatic error analysis; it was used extensively by Wilkinson on the ACE machine. Various forms of automatic error analysis have been developed. In this chapter we describe in detail the use of direct search optimization for investigating questions about the stability and accuracy of algorithms. We also describe interval analysis and survey other forms of automatic error analysis.
24.1. Exploiting Direct Search Optimization Is Algorithm X numerically stable? How large can the growth factor be for Gaussian elimination (GE) with pivoting strategy P? By how much can condition estimator C underestimate the condition number of a matrix? These types of questions are common, as we have seen in this book. Usually, we attempt to answer such questions by a combination of theoretical analysis and numerical experiments with random and nonrandom data. But a third approach can be a valuable supplement to the first two: phrase the question as an optimization problem and apply a direct search method. A direct search method for the problem (24.1) is a numerical method that attempts to locate a maximizing point using function values only and does not use or approximate derivatives of f. Such methods are usually based on heuristics that do not involve assumptions about the function f. Various direct search methods have been developed; for surveys see Powell [838, 1970] and Swarm [978, 1972], [979, 1974]. Most of these methods were developed in the 1960s, in the early years of numerical optimization. For problems in which f is smooth, direct search methods have largely been supplanted by more sophisticated optimization methods that use derivatives (such as quasi-Newton methods and conjugate gradient methods), but they continue to find use in applications where f is not differentiable, or even not continuous. These applications range from chemical analysis [881, 1977], where direct search methods have found considerable use, to the determination of drug doses in the treatment of cancer [93, 1991]; in both applications the evaluation of f is affected by experimental errors. Lack of smoothness of f, and the difficulty of obtaining derivatives when they exist, are characteristic of the optimization problems we consider here. The use of direct search can be illustrated with the example of the growth
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factor for GE on A
where the are the intermediate elements generated during the elimination. We know from 9.2 that the growth factor governs the stability of GE, so for a given pivoting strategy we would like to know how big ρ n (A) can be. To obtain an optimization problem of the form (24.1) we let x = vet(A) and we define f(x) := ρn(A). Then we wish to determine
Suppose, first, that no pivoting is done. Then f is defined and continuous at all points where the elimination does not break down, and it is differentiable except at points where there is a tie for the maximum in the numerator or denominator of the expression defining ρn(A). We took n = 4 and applied the direct search maximizer MDS (described in 24.2) to f(x), starting with the identity matrix A = I4. After 11 iterations and 433 function evaluations, the maximizer converged18 , having located the matrix19
for which ρ4(B) = 1.23 x 105. (The large growth is a consequence of the submatrix B(1:3, 1:3) being ill conditioned; B itself is well conditioned.) Thus the optimizer readily shows that ρn(A) can be very large for GE without pivoting. Next, consider GE with partial pivoting (GEPP). Because the elimination cannot break down, f is now defined everywhere, but it is usually discontinuous when there is a tie in the choice of pivot element, because then an arbitrarily small change in A can alter the pivot sequence. We applied the maximizer MDS to f, this time starting with the orthogonal matrix20 A with aij = (cf. (9.11)), for which ρ4(A) = 2.32. After 29 iterations and 1169 function evaluations the maximizer converged to a matrix B with ρ 4(B) = 5.86. We used this matrix to start the maximizer 18 In the optimizations of this section we used the convergence tests described in 24.2 with tol = 10-3. There is no guarantee that when convergence is achieved it is to a local maximum; see 24.2. 19 A11 numbers quoted are rounded to the number of significant figures shown. 20 This matrix is orthog (n, 2) from the Test Matrix Toolbox; see Appendix E.
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AD (described in 24.2); it took 5 iterations and 403 function evaluations to converge to the matrix
for which ρ4(C) = 7.939. This is one of the matrices described in Theorem 9.6, modulo the convergence tolerance. These examples, and others presented below, illustrate the following attractions of using direct search methods to investigate the stability of a numerical computation. (1) The simplest possible formulation of optimization problem is often sufficient to yield useful results. Derivatives are not needed, and direct search methods tend to be insensitive to lack of smoothness in the objective function f. Unboundedness of f is a favorable property-direct search methods usually quickly locate large values of f. (2) Good progress can often be made from simple starting values, such as an identity matrix. However, prior knowledge of the problem may provide a good starting value that can be substantially improved (as in the partial pivoting example). (3) Usually it is the global maximum of f in (24.1) that is desired (although it is often sufficient to know that f can exceed a specified value). When a direct search method converges it will, in general, at best have located a local maximum-and in practice the maximizer may simply have stagnated, particularly if a slack convergence tolerance is used. However, further progress can often be made by restarting the same (or a different) maximizer, as in the partial pivoting example. This is because for methods that employ a simplex (such as the MDS method), the behaviour of the method starting at x0 is determined not just by x0. but also by the n + 1 vectors in the initial simplex constructed at x0 . (4) The numerical information revealed by direct search provides a starting point for further theoretical analysis. For example, the GE experiments above strongly suggest the (well known) results that ρ n(A) is unbounded without pivoting and bounded by 2n–1 for partial pivoting, and inspection of the numerical data suggests the methods of proof. When applied to smooth problems the main disadvantages of direct search methods are that they have at best a linear rate of convergence and they are unable to determine the nature of the point at which they terminate (since derivatives are not calculated). These disadvantages are less significant for the problems we consider, where it is not necessary to locate a maximum to high accuracy and objective functions are usually nonsmooth. (Note that these disadvantages are not necessarily shared by methods that implicitly or
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explicitly estimate derivatives using function values, such as methods based on conjugate directions, for which see Powell [838, 1970], [839, 1975]; however, these are not normally regarded as direct search methods.) A final attraction of direct search is that it can be used to test the correctness of an implementation of a stable algorithm. The software in question can be used in its original form and does not have to be translated into some other representation.
24.2. Direct Search Methods For several years I have been using MATLAB implementations of three direct search methods. The first is the alternating directions (AD) method (also known as the coordinate search method). Given a starting value x it attempts to solve the problem (24.1) by repeatedly maximizing over each coordinate direction in turn: repeat % One iteration comprises a loop over all components of x. for i = l:n find α such that f(x + αei ) is maximized (line search) Set x x + αe i end until converged AD is one of the simplest of all optimization methods and the fundamental weakness that it ignores any interactions between the variables is well known. Despite the poor reputation of AD we have found that it can perform well on the types of problems considered here. In our MATLAB implementation of AD the line search is done using a crude scheme that begins by evaluating f(x + hei ) with h = 10 -4xi (or h = 1 0 - 4 m a x if xi = 0); if f(x + he i ) < f(x) then the sign of h is reversed. Then if f(x + he i ) > f(x), h is doubled at most 25 times until no further increase in f is obtained. Our convergence test checks for a sufficient relative increase in f between one iteration and the next: convergence is declared when f k – f k - 1 < t o l|fk-1|,
(24.2)
where fk is the highest function value at the end of the k th iteration. The AD method has the very modest storage requirement of just a single n-vector. The second method is the multidirectional search method (MDS) of Dennis and Torczon [1008, 1989], [1009, 1991], [301, 1991]. This method employs a simplex, which is defined by n + 1 vectors One iteration in the case n = 2 is represented pictorially in Figure 24.1, and may be explained as follows.
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Figure 24.1. The possible steps in one iteration of the MDS method when n = 2.
The initial simplex is {v0, v1, v2} and it is assumed that f(v0) = max i f(vi ). The purpose of an iteration is to produce a new simplex at one of whose vertices f exceeds f(v0 ). In the first step the vertices v1 and v2 are reflected about v0 . along the lines joining them to v0 , yielding rl and r2 and the reflected simplex {v 0 ,r 1 ,r 2 }. If this reflection step is successful, that is, if maxi f(ri ) > f(v0), then the edges from v0 to ri are doubled in length to give an expanded simplex {v0, el, e2 }. The original simplex is then replaced by {v0, el, e2 } if maxi f(ei ) > maxi f(ri ), and otherwise by {v0, r1, r2 }. If the reflection step is unsuccessful then the edges v0 – vi of the original simplex are shrunk to half their length to give the contracted simplex {v0, c1, c2}. This becomes the new simplex if max i f(ci ) > maxi f(vi ), in which case the current iteration is complete; otherwise the algorithm jumps back to the reflection step, now working with the contracted simplex. For further details of the MDS method see Dennis and Torczon [301, 1991] and Torczon [1008, 1989], [1009, 1991]. The MDS method requires at least 2n independent function evaluations per iteration, which makes it very suitable for parallel implementation. Generalizations of the MDS method that are even more suitable for parallel computation are described in [301, 1991] and [1010, 1992]. The MDS method requires O(n2) elements of storage for the simplices, but this can be reduced to O(n) (at the cost of extra bookkeeping) if an appropriate choice of initial simplex is made [301, 1991]. Unlike most direct search methods, the MDS method possesses some convergence theory. Torczon [1009, 1991] shows that if the level set of j at
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is compact and f is continuously differentiable on this level set then a subsequence of the points (where k denotes the iteration index) converges to a stationary point of f. Moreover, she gives an extension of this result that requires only continuity of f and guarantees convergence to either a stationary point off or a point where f is not continuously differentiable. Our implementation of the MDS method provides two possible starting simplices, both of which include the starting point x0: a regular one (all sides of equal length) and a right-angled one based on the coordinate axes, both as described by Torczon in [1008, 1989]. The scaling is such that each edge of the regular simplex, or each edge of the right-angled simplex that is joined to x0 , has length Also as in [1008, 1989], the main termination test halts the computation when the relative size of the simplex is no larger than a tolerance toll that is, when (24.3) Unless otherwise stated, we used tol = 10-3 in (24.2) and (24.3) in all our experiments. The third method that we have used is the Nelder–Mead direct search method [787, 1965], [303, 1987], which also employs a simplex but which is fundamentally different from the MDS method. We omit a description since the method is described in textbooks (see, for example, Gill, Murray, and Wright [447, 1981, §4.2.2], or Press, Teukolsky, Vetterling, and Flannery [842, 1992, §10.4]). Our limited experiments with the Nelder-Mead method indicate that while it can sometimes out-perform the MDS method, the MDS method is generally superior for our purposes. No convergence results of the form described above for the MDS method are known for the Nelder-Mead method. For general results on the convergence of “pattern search” methods, see Torczon [1011, 1993]. The AD and MDS methods are pattern search methods, but the Nelder–Mead method is not. It is interesting to note that the MDS method, the Nelder-Mead method, and our particular implementation of the AD method do not exploit the numerical values of f: their only use of f is to compare two function values to see which is the larger! Our MATLAB implementations of the AD, MDS, and Nelder-Mead direct search methods are in the Test Matrix Toolbox, described in Appendix E.
24.3. Examples of Direct Search In this section we give examples of the use of direct search to investigate the behaviour of numerical algorithms.
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480 24.3.1. Condition Estimation
We have experimented with MATLAB implementations of two matrix condition number estimators. RCOND is the LINPACK estimator, described in §14.4, as implemented in the built-in MATLAB function rcond. LACON is the LAPACK condition estimator, as given in Algorithm 14.4 and implemented in MATLAB’S condest. Both estimators compute a lower bound for k1(A) by estimating ||A–1 ||1 (||A||1 is computed explicitly as the maximum column sum). To put the problem in the form of (24.1), we define x = vec(A), A and
where est (A) < k1 (A) is the condition estimate. We note that, since the algorithms underlying RCOND and LACON contain tests and branches, there are matrices A for which an arbitrarily small change in A can completely change the condition estimate; hence for both algorithms f has points of discontinuity. We applied the MDS maximizer to RCOND starting at the 5 x 5 Hilbert matrix. After 67 iterations and 4026 function evaluations the maximizer had located a matrix for which f(x) = 226.9. We then started the NelderMead method from this matrix; after another 4947 function evaluations it had reached the matrix (shown to 5 significant figures)
for which K1(A)
= 3.38 x 10 5 , est(A) = 1.65 x 101,
This example is interesting because the matrix is well scaled, while the parametrized counterexamples of Cline and Rew [217, 1983] all become badly scaled when the parameter is chosen to make the condition estimate poor. For LACON we took as starting matrix the 4 x 4 version of then x n matrix with a ij = cos((i – 1)(j – 1)π /( n – 1)) (this is a Chebyshev–Vandermonde matrix, as used in §21.3.3, and is orthog(n, - 1) in the Test Matrix Toolbox). After 11 iterations and 1001 function evaluations the AD maximizer had determined a (well-scaled) matrix A for which k1(A)
= 2.94 x 105,
est(A) = 4.81,
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With relatively little effort on our part (most of the effort was spent experimenting with different starting matrices), the maximizers have discovered examples where both condition estimators fail to achieve their objective of producing an estimate correct to within an order of magnitude. The value of direct search maximization in this context is clear: it can readily demonstrate the fallibility of a condition estimator—a task that can be extremely difficult to accomplish using theoretical analysis or tests with random matrices. Moreover, the numerical examples obtained from direct search may provide a starting point for the construction of parametrized theoretical ones, or for the improvement of a condition estimation algorithm. In addition to measuring the quality of a single algorithm, direct search can be used to compare two competing algorithms to investigate whether one algorithm performs uniformly better than the other. We applied the MDS maximizer to the function
where estL(A) and estR(A) are the condition estimates from LACON and RCOND, respectively. If f(x) > 1 then LACON has produced a larger lower bound for k1 (A) than RCOND. Starting with a random 5 x 5 matrix the Nelder–Mead maximizer produced after 1788 function evaluations a matrix A for which estL(A) = k1(A) and f(x) = 1675.4. With f defined as f(x) = estR(A)/estL(A), and starting with I4, after 6065 function evaluations the MDS maximizer produced a matrix for which f(x) = 120.8. This experiment shows that neither estimator is uniformly superior to the other. This conclusion would be onerous to reach by theoretical analysis of the algorithms. 24.3.2. Fast Matrix Inversion We recall Strassen’s inversion method from Problem 22.8: for
it uses the formulae P1 = P3 = P1 A12 , P5 = P4 – A22,
P2 = A21 P1 , P4 = A21 P3 , P6 = P5–1,
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where each of the matrix products is formed using Strassen’s fast matrix multiplication method. Strassen’s inversion method is clearly unstable for general A, because the method breaks down if A11 is singular. Indeed Strassen’s inversion method has been implemented on a Cray-2 by Bailey and Ferguson [41, 1988] and tested for n < 2048, and these authors observe empirically that the method has poor numerical stability. Direct search can be used to gain insight into the numerical stability. With x = vet( A )
define the stability measure
(24.4)
where is the inverse of A computed using Strassen’s inversion method. This definition of f is appropriate because, as shown in Chapter 13, for most conventional matrix inversion methods either the left residual XA – I or the right residual AX – I is guaranteed to have norm of order u||X|| ||A||. To treat Strassen’s inversion method as favorably as possible we use just one level of recursion; thus P1 and P6 are computed using GEPP but the multiplications are done with Strassen’s method. We applied the MDS maximizer, with tol = 10-9 in (24.3), starting with the 4 x 4 Vandermonde matrix whose (i,j) element is ((j – 1)/3) i–1. After 34 iterations the maximizer had converged with f = 0.838, which represents complete instability. The corresponding matrix A is well conditioned, with k2(A) = 82.4. For comparison, the value of f when A is inverted using Strassen’s method with conventional multiplication is f = 6.90 x 10–2; this confirms that the instability is not due to the use of fast multiplication techniques—it is inherent in the inversion formulae. If A is a symmetric positive definite matrix then its leading principal submatrices are no more ill conditioned than the matrix itself, so we might expect Strassen’s inversion method to be stable for such matrices. To investigate this possibility we carried out the same maximization as before, except we enforced positive definiteness as follows: when the maximizer generates a vector x = vec(B), A in (24.4) is defined as A = BTB. Starting with a 4 x 4 random matrix A with k2(A) = 6.71 x 107, the maximization yielded the value f = 3.32 x 10 -8 after 15 iterations, and the corresponding value of f when conventional multiplication is used is f = 6.61 x 10 –11 (the “maximizing” matrix A has condition number k2(A) = 3.58 x 109). The conclusion from these experiments is that Strassen’s inversion method cannot be guaranteed to produce a small left or right residual even when A is symmetric positive definite and conventional multiplication is used. Hence the method must be regarded as being fundamentally unstable.
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24.3.3. Solving a Cubic Explicit formulae can be obtained for the roots of a cubic equation using techniques associated with the 16th century mathematicians del Ferro, Cardano, Tartaglia, and Vieta [140, 1968], [332, 1990]. The following development is based on Birkhoff and Mac Lane [102, 1977, §5.5]. Any nondegenerate cubic equation can be put in the form x3 + ax2 + bx + c = O by dividing through by the leading coefficient. We will assume that the coefficients are real. The change of variable x = y – a/3 eliminates the quadratic term: y3 + py + q = 0, Then Vieta’s substitution y = w – p/(3w ) yields
and hence a quadratic equation in w3: (W3)2 + qw3 — p3 /27 = 0. Hence (24.5) For either choice of sign, the three cube roots for w yield the roots of the original cubic, on transforming back from w to y to Z. Are these formulae for solving a cubic numerically stable? Direct search provides an easy way to investigate. The variables are the three coefficients a, b, c and for the objective function we take an approximation to the relative 3 . We compute the “exact” roots z userror of the computed roots ing MATLAB’s roots function (which uses the QR algorithm to compute the eigenvalues of the companion matrix21 ) and then our relative error measure is where we minimize over all six permutations Π. First, we arbitrarily take the "+" square root in (24.5). With almost any starting vector of coefficients, the MDS maximizer rapidly identifies coefficients for which the computed roots are very inaccurate. For example, starting with [1, 1, 1]T we are lead to the vector [ a b c] T = [1.732 1 1.2704]T , for which the computed and “exact” roots are
21 Edelman and Murakami [346, 1995] and Toh and Trefethen [1007, 1994] analyse the stability of this method of finding polynomial roots; the method is stable.
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When the cubic is evaluated at the computed roots the results are of order 10–2, whereas they are of order 10–15 for z. Since the roots are well separated the problem of computing them is not ill conditioned, so we can be sure that z is an accurate approximation to the exact roots. The conclusion is that the formulae, as programmed, are numerically unstable. Recalling the discussion for a quadratic equation in §1.8, a likely reason for the observed instability is cancellation in (24.5). Instead of always taking the “+” sign, we therefore take (24.6) When the argument of the square root is nonnegative, this formula suffers no cancellation in the subtraction; the same is true when the argument of the square root is negative, because then the square root is pure imaginary. With the use of (24.6), we were unable to locate instability using the objective function described above. However, an alternative objective function can be derived as follows. It is reasonable to ask that the computed roots be the exact roots of a slightly perturbed cubic. Thus each computed root should be a root of
where is of the order of the unit roundoff. Notice that we are allowing the leading coefficient of unity to be perturbed. Denoting the unperturbed cubic by f, we find that this condition implies that . (24.7) is of order u. We therefore take the quantity in (24.7) as the function to be maximized. On applying the MDS maximizer with starting vector [1, 1, 1] T , we obtain after 10 iterations an objective function value of 1.2 x 10–11. The cubic coefficient vector is (to three significant figures) [a b c] T = [-5.89 x 102 3.15 x 102 -1.36 x 101], and the computed roots are (ignoring tiny, spurious imaginary parts) = [4.75 x 10 -2 4.87 x 10 -1 5.89 x 10 2 ]. The value of the objective function corresponding to the “exact” roots (computed as described above) is of order 10-16 (and the value of the previous relative error objective function for the computed roots is of order 10–14 ). The conclusion is that even using (24.6) the formulae for the roots are numerically unstable. However, further theoretical analysis is needed to understand the stability fully; see Problem 24.3.
24.4 INTERVAL ANALYSIS
485
24.4. Interval Analysis Interval analysis has been an area of extensive research since the 1960s, and it had been considered earlier by Turing and Wilkinson in the 1940s [1099, 1980, p. 104]. As the name suggests, the idea is to perform arithmetic on intervals [a,b] (b > a). The aim is to provide as the solution to a problem an interval in which the desired result is guaranteed to lie, which may be particularly appropriate if the initial data is uncertain (and hence can be thought of as an interval). For the elementary operations, arithmetic on intervals is defined by
and the results are given directly by the formulae [a,b] + [c,d] = [a + c, b + d], [a,b] - [c,d] = [a - d, b - c], [a,b] * [c,d] = [min (ac, ab, bc, bd), max (ac, ab, bc, bd)], [a,b] / [c,d] = [a,b] * [1/d, 1/c ], We will use the notation [x] for an interval [xl,x2] and we define width(x) := x2 – x1 . In floating point arithmetic, an interval containing fl( [x] op [y]) is obtained by rounding computations on the left endpoint to and those on the right endpoint to (both these rounding modes are supported in IEEE arithmetic). The success of an interval analysis depends on whether an answer is produced at all (an interval algorithm will break down with division by zero if it attempts to divide by an interval containing zero), and on the width of the interval answer. A one-line program that prints would be correct, robust, and fast, but useless. Interval analysis is controversial because, in general, narrow intervals cannot be guaranteed. One reason is that when dependencies occur in a calculation, in the sense that a variable appears more than once, final interval lengths can be pessimistic. For example, if [x ] = [1,2] then [ x] - [x] = [-1, 1], [x]/[x] = [1/2, 2], whereas the optimal intervals for these calculations are, respectively, [0, O] and [1, 1]. These calculations can be interpreted as saying that there is no additive or multiplicative inverse in interval arithmetic. An example of an algorithm for which interval arithmetic can be ineffective is GEPP, which in general gives very pessimistic error bounds and is unstable in interval arithmetic even for a well-conditioned matrix [798, 1977]. The
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basic problem is that the interval sizes grow exponentially. For example, in the 2 x 2 reduction
if
then width([ y ] – [x])
width([ x] – [x]) = 2 width([x]).
This type of growth is very likely to happen, unlike the superficially similar phenomenon of element growth in standard GEPP. The poor interval bounds are entirely analogous to the pessimistic results returned by early forward error analyses of GEPP (see §9.6). Nickel [798, 1977] states that “The interval Gauss elimination method is one of the most unfavorable cases of interval computation . . . nearly all other numerical methods give much better results if transformed to interval methods”. Interval GE is effective, however, for certain special classes of matrices, such as M-matrices. As already mentioned, there is a large body of literature on interval arithmetic, though, as Skeel notes (in an article that advocates interval arithmetic), “elaborate formalisms and abstruse notation make much of the literature impenetrable to all but the most determined outsiders” [922, 1989]. Good sources of information include various conference proceedings and the journal Computing. The earliest book is by Moore [778, 1966], whose later book [779, 1979] is one of the best references on the subject. Nickel [798, 1977] gives an easy to read summary of research up to the mid 1970s. A more recent reference is Alefeld and Herzberger [10, 1983]. Yohe [1120, 1979] describes a Fortran 66 package for performing interval arithmetic in which machine-specific features are confined to a few modules. It is designed to work with a precompiled for Fortran called Augment [254, 1979], which allows the user to write programs as though Fortran had extended data types—in this case an INTERVAL type. A version of the package that allows arbitrary precision interval arithmetic by incorporating Brent’s multiple precision arithmetic package (see §25.9) is described in [1121, 1980]. In [1119, 1979], Yohe describes general principles of implementing nonstandard arithmetic in software, with particular reference to interval arithmetic. Kulisch and Miranker have proposed endowing a computer arithmetic with a super-accurate inner product, that is, a facility to compute an exactly rounded inner product for any two vectors of numbers at the working precision [678, 1981], [679, 1983], [680, 1986]. This idea has been implemented in the package ACRITH from IBM, which employs interval arithmetic [124, 1985]. Anomalies in early versions of ACRITH are described by Kahan and LeBlanc [639, 1985] and Jansen and Weidner [612, 1986]. For a readable discussion of interval arithmetic and the super-accurate inner product, see Rail [858, 1991]. Cogent arguments against adopting a super-accurate inner product are given by Demmel [284, 1991].
24.5 O THER W ORK
487
Fortran and Pascal compilers (Fortran SC and Pascal-XSC) and a C++ class library that support a super-accurate inner product and interval arithmetic have been developed jointly by researchers at IBM and the University of Karlsruhe [125, 1987], [661, 1993], [662, 1992]. A toolbox of routines written in Pascal-XSC for solving basic numerical problems and verifying the results is presented by Hammer, Hocks, Kulisch, and Ratz [500, 1993]. Finally, we explain why any attempt to compute highly accurate answers using interval arithmetic of a fixed precision (possibly combined with a superaccurate inner product ) cannot always succeed. Suppose we have a sequence of problems to solve, where the output of one is the input to the next: xi+1 = f i (x i ), i = 1:n, for smooth functions fi : Suppose x 1 is known exactly (interval width zero). Working in finite precision interval arithmetic, we obtain an interval [x2 – α2, x2 + β 2] containing x2. This is used as input to the computation of f 2 . Even under the favorable assumption of no rounding errors in the evaluation of f2, we obtain an interval answer whose width must be of order
(the interval could be much bigger than f 2 (x2 ) – f 2 (x2 ± ε 2), depending on the algorithm used to evaluate f2). In other words, the width of the interval containing x3 is roughly proportional to the condition number of f 2 . When the output of the f2 computation is fed into f3 the interval width is multiplied by f´3 The width of the final interval containing xn+l is proportional to the product of the condition numbers of all the functions f i and if there are enough functions, or if they are conditioned badly enough, the final interval will provide no useful information The only way to avoid such a failure is to use variable precision arithmetic or to reformulate the problem to escape the product of the condition numbers of the fi .
24.5. Other Work In this section we outline other work on automatic error analysis. In the 1970s, Miller and his co-authors developed methods and software for automatically searching for numerical instability in algebraic processes [757, 1975], [760, 1978], [762, 1980]. In [757, 1975] Miller defines a quantity σ ( d ) that bounds, to first order, the sensitivity of an algorithm to perturbations in the data d and in the intermediate quantities that the algorithm generates. He then defines the forward stability measure ρ(d) = σ(d)/k(d), where k(d) is a condition number for the problem under consideration. The algorithms to be analysed are required to contain no loops or conditional branches and are presented to Miller’s Fortran software in a special numeric encoding. The software automatically computes the partial derivatives needed to evaluate
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ρ(d), and attempts to maximize ρ using the method of alternating directions. Miller gives several examples illustrating the scope of his software; he shows, for example, that it can identify the instability of the classical Gram-Schmidt method for orthogonalizing a set of vectors. In [760, 1978], [761, 1978] Miller and Spooner extend the work in [757, 1975] in several ways. The algorithm to be analysed is expressed in a Fortranlike language that allows for-loops but not logical tests. The definition of p is generalized and a method of computing it is developed that involves solving a generalized eigenvalue problem. The book by Miller and Wrathall [762, 1980] gives a thorough development of the work of [760, 1978], including a description of the graph theory techniques used to compute the partial derivatives, and it provides further examples of the use of the software. The potential of Miller and Spooner’s software for exposing numerical instability is clearly demonstrated by the case studies in these references, yet the software has apparently not been widely used. This is probably largely due to the inability of the software to analyse algorithms expressed in Fortran, or any other standard language. A different approach to algorithm analysis is taken by Larson and Sameh [688, 1978], [689, 1980], and implemented in software by Larson, Pasternak, and Wisniewski [687, 1983]. Here, errors are measured in a relative rather than an absolute sense, and the stability is analysed at fixed data instead of attempting to maximize instability over all data; however, the analysis is still linearized. The idea of applying automatic differentiation to a computational graph to obtain estimates for the forward error in an algorithm is found not only in the references cited above, but also in the automatic differentiation literature; see Rail [857, 1981], Iri [606, 1991], and Kubota [675, 1991], for example. Hull [589, 1979] discusses the possibility of applying program verification techniques to computations that are subject to rounding errors, with the aim of proving a program “correct”. Difficulties include deciding what “correct” means and formulating appropriate assertions. Although he reports some progress, Hull concludes that “there is not a great deal to be gained by trying to apply the techniques of program verification to programs for numerical calculations”. Bliss, Brunet, and Gallopoulos [126, 1992] develop Fortran preprocessor tools for implementing the local relative error approach of Larson and Sameh. Their tools also implement a statistical technique of Brunet and Chatelin [152, 1989], [202, 1990] in which the result of every floating point operation is randomly perturbed and statistical methods are used to measure the effect on the output of the algorithm. Rowan [882, 1990] develops another way to search for numerical instability. For an algorithm with data d he maximizes S(d) = e(d)/ k (d) using a direct search maximizer he has developed called the subplex method (which
24.6 N OTES
AND
R EFERENCES
489
is based on the Nelder–Mead simplex method). Here, e(d) = yacc - is an approximation to the forward error in the computed solution where yacc is a more accurate estimate of the true solution than and the condition number k(d) is estimated using finite difference approximations. The quantity S(d) is a lower bound on the backward error of the algorithm at d. Fortran software given in [882, 1990] implements this “functional stability analysis”. The software takes as input two user-supplied Fortran subprograms; one implements the algorithm to be tested in single precision, and the other provides a more accurate solution, typically by executing the same algorithm in double precision. The examples in [882, 1990] show that Rowan’s software is capable of detecting numerical instability in a wide variety of numerical algorithms. A technique called “significance arithmetic” was studied by Ashenhurst, Metropolis, and others in the 1960s. It involves performing arithmetic on unnormalized numbers in such a way that the number of significant digits in the answers provides estimates of the accuracy of those answers. Significance arithmetic is therefore a form of automatic error analysis. For more details see, for example, Ashenhurst and Metropolis [31, 1965], [750, 1977] and Sterbenz [938, 1974, §7.2]; Sterbenz explains several drawbacks of the technique. Stoutemyer [958, 1977] describes the use of the symbolic manipulation package REDUCE to analyse the propagation of data and rounding errors in numerical algorithms. Finally, we note that running error analysis (see §3.3) is a form of automatic error analysis and is an attractive alternative to interval arithmetic as a means of computing a posteriori error bounds for almost any numerical algorithm.
24.6. Notes and References Sections 24.1 – 24.3.2 and §24.5 are based on Higham [555, 1993]. The Release Notes for MATLAB 4.1 state that “This release of MATLAB fixes a bug in the rcond function. Previously, rcond returned a larger than expected estimate for some matrices . . . rcond now returns an estimate that matches the value returned by the Fortran LINPACK library.” In the direct search experiments in [555, 1993] we used MATLAB 3.5, and we found it much easier to generate counterexamples to rcond than we do now with MATLAB 4.2. It seems that the maximizations in [555, 1993] were not only defeating the algorithm underlying rcond, but also, unbeknown to us, exploiting a bug in the implemental ion of the function. The conclusions of [555, 1993] are unaffected, however. Another way to solve a cubic is to use Newton’s method to find a real zero, and then to find the other two zeros by solving a quadratic equation.
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In a detailed investigation, Kahan [631, 1986] finds this iterative method to be preferable to (sophisticated) use of the explicit formulae. Other useful references on the numerical computation of roots of a cubic are Lanczos [686, 1956, Chap. 1], Press, Teukolsky, Vetterling, and Flannery [842, 1992, §5.6], and Uspensky [1038, 1948].
Problems 24.1. Let A and let and be the computed orthogonal QR factors produced by the classical and modified Gram-Schmidt methods, respectively. Use direct search to maximize the functions
In order to keep k2(A) small, try maximizing fi (A) – θ max( k2(A) – µ, 0), where θ is a large positive constant and µ is an upper bound on the acceptable condition numbers. 24.2. It is well known that if then
is nonsingular and vTA–1 u
–1
This is known as the Sherman-Morrison formula. For a history and generalizations see Henderson and Searle [512, 1981]. A natural question is whether this formula provides a stable way to solve a rank-1 perturbed linear system. That is, is the following algorithm stable? % Solve Cx: = (A + uvT)x = b. Solve Ay = b for y. Solve Az = u for z. x = y - (v T y)(1 + v T z) - 1 z (a) Investigate the stability using direct search. Let both linear systems with coefficient matrix A be solved by GEPP. Take A, u, and v as the data and let the function to be maximized be the normwise backward error η C,b in the norm. (b) (RESEARCH PROBLEM) Obtain backward and forward error bounds for the method (for some existing analysis see Yip [1118, 1986]). 24.3. (RESEARCH PROBLEM) Investigate the stability of the formulae of $24.3.3 for computing the roots of a cubic.
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Chapter 25
Software Issues in Floating Point Arithmetic
The first American Venus probe was lost due to a program fault caused by the inadvertent substitution of a statement of the form DO 3 I = 1.3 for one of the form DO 3 I = 1,3.
— JIM HORNING, Note on Program Reliability 2 2 ( 1 9 7 9 ) Numerical subroutines should deliver results that satisfy simple, useful mathematical laws whenever possible. — DONALD E. KNUTH, The Art of Computer Programming, Volume 2, Seminumerical Algorithms (1981) No method of solving a computational problem is really available to a user until it is completely described in an algebraic computing language and made completely reliable. Before that, there are indeterminate aspects in any algorithm. — GEORGE E. FORSYTHE, Today’s Computational Methods of Linear Algebra (1967) The extended precision calculation of pi has substantial application as a test of the “global integrity” of a supercomputer . . . Such calculations . . . are apparently now used routinely to check supercomputers before they leave the factoty. A large-scale calculation of pi is entirely unforgiving; it soaks into all parts of the machine and a single bit awry leaves detectable consequences. — J. M. BORWEIN, P. B. BORWEIN, and D. H. BAILEY, Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi (1989) 22
Quoted, with further details, in Tropp [1021, 1984].
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IN
In this chapter we discuss some miscellaneous aspects of floating point arithmetic that have an impact on software development.
25.1. Exploiting IEEE Arithmetic IEEE standard 754 and 854 arithmetic can be exploited in several ways in numerical software, provided that proper access to the arithmetic’s features is available in the programming environment. Unfortunately, although most commercially significant floating point processors at least nearly conform to the IEEE standards, language standards and compilers generally provide poor support (exceptions include the Standard Apple Numerics Environment (SANE) [233, 1988], Apple’s PowerPC numerics environment [234, 1994], and Sun’s SPARCstation compilers [969, 1992], [970, 1992]). We give four examples to show the benefits of IEEE arithmetic on software. Suppose we wish to evaluate the dual of the vector p- n o r m that is, the g-norm, where p –l + q –1 = 1. In MATLAB notation we simply write norm (x, 1/(1-l/p)), and the extreme cases p = 1 and correctly yield q = 1/(1 – 1/p) = and 1 in IEEE arithmetic. (Note that the formula q = p/ (p – 1) would not work, because inf/inf evaluates to a NaN.) The following code finds the biggest element in an array a (1: n), and elegantly solves the problem of choosing a “sufficiently negative number” with which to initialize the variable max: max = –inf for j = 1:n if aj > max max = aj end end Any unknown or missing elements of the array a could be assigned NaNs (a(j) = NaN) and the code would (presumably) still work, since a NaN compares as unordered with everything. Consider the following continued fraction [630, 1981]: 3
r(x) = 7 –
,
1
x - 2 –
10
x - 7 + x - 2
2 x - 3
which is plotted over the range [0,4] in Figure 25.1. Another way to write the
25.1 E XPLOITING IEEE A RITHMETIC
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Figure 25.1. Rational function r. rational function r(x) is in the form
in which the polynomials in the numerator and denominator are written in the form in which they would be evaluated by Homer’s rule. Examining these two representations of r, we see that the continued fraction requires less arithmetic operations to evaluate (assuming it is evaluated in the obvious “bottom up” fashion) but it incurs division by zero at the points x = 1:4, even though r is well behaved at these points. However, in IEEE arithmetic r evaluates correctly at these points because of the rules of infinity arithmetic. For x = 1077, the rational form suffers overflow, while the continued fraction evaluates correctly to 7.0; indeed, in IEEE arithmetic the continued fraction is immune to overflow. Figure 25.2 shows the relative errors made in evaluating r in double precision on an equally spaced grid of points on the range [0,4] (many of the errors for the continued fraction are zero, hence the gaps in the error curve); clearly, the continued fraction produces the more accurate function values. The conclusion is that in IEEE arithmetic the continued fraction representation is the preferred one for evaluating r. The exception handling provided by IEEE arithmetic can be used to simplify and speed up certain types of software, provided that the programming environment properly supports these facilities. Recall that the exceptional
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Figure 25.2. Error in evaluating rational function r. Solid line: continued fraction, dotted line: usual rational form.
operations underflow, overflow, divide by zero, invalid operation, and inexact deliver a result, set a flag, and continue. The flags (one for each type of exception) are “sticky”, remaining set until explicitly cleared, and implementations of the IEEE standard are required to provide a means to read and write the flags. Unfortunately, compiler writers have been slow to make exception handling, and some of IEEE arithmetic’s other unusual features, available to the programmer. Kahan has commented that “the fault lies partly with the IEEE standard, which neglected to spell out standard names for the exceptions, their flags, or even As an example of how exception handling can be exploited, consider the use of the LAPACK norm estimator (Algorithm 14.4) to estimate ||A–1 ||1 . The algorithm requires the solution of linear systems of the form Ax = b and ATy = c, which is typically done with the aid of an LU factorization PA = LU. Solution of the resulting triangular systems is prone to both overflow and division by zero. In LAPACK, triangular systems are solved in this context not with the level-2 BLAS routine xTRSV but with a routine xLATRS that contains elaborate tests and scalings to avoid overflow and division by zero [18, 1991]. In an IEEE arithmetic environment a simpler and potentially faster approach is possible: solve the triangular systems with xTRSV and after computing each solution check the flags to see whether any exceptions occurred. Provided that some tests are added to take into account overflow due solely to ||A||1 being
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tiny, the occurrence of an exception allows the conclusion that A is singular to working precision; see Demmel and Li [298, 1994] for the details. For the condition estimator that uses exception handling to be uniformly faster than the code with scaling it is necessary that arithmetic with NaNs, infinities , and subnormal numbers be performed at the same speed as conventional arithmetic. While this requirement is often satisfied, there are machines for which arithmetic on NaNs, infinities, and subnormal numbers is between one and three orders of magnitude slower than conventional arithmetic [298, 1994, Table 2]. Demmel and Li compared the LAPACK condition estimation routines xxxCON with modified versions containing exception handling and found the latter to be up to 6 times faster and up to 13 times slower, depending on the machine and the matrix. Appendices to the IEEE standards 754 and 854 recommend 10 auxiliary functions and predicates to support portability of programs across different IEEE systems. These include nextafter(x,y), which returns the next floating point number to x in the direction of y, and scalb(x,n), which returns x x β n without explicitly computing β n, where n is an integer and β the base. Not all implementations of IEEE arithmetic support these functions. Portable C versions of six of the functions are presented by Cody and Coonen [226, 1993].
25.2. Subtleties of Floating Point Arithmetic The difference x – y of two machine numbers can evaluate to zero even when because of underflow. This makes a test such as
f = f / (x - y) end unreliable. However, in a system that supports gradual underflow (such as IEEE arithmetic) x – y always evaluates as nonzero when x y, as is easy to verify. On several models of Cray computer (Cray 1, 2, X-MP, Y-MP, and C90) this code could fail for another reason: they compute f/(x – y) as f * (1/(x – y)) and 1/(x – y) could overflow. It is a general principle that one should not test two floating point numbers for equality, but rather use a test such as if |x – y| < tol (there are exceptions, such as Algorithm 2 in §1.14.1). Of course, skill may be required to choose an appropriate value for tel. A test of this form would correctly avoid the division in the example above when underflow occurs.
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Table 25.1. Results from Cholesky factorization. Bits 128 64 64 64 64 64
u le-29 le-17 le-16 le-16 le-15 le-15
Computer Cray 2 IBM 3090 Convex 220 IRIS Cray 2 Cray Y-MP
Displacement .447440341 .447440344 .44744033 9 .44744033 9 .447440303 .447436106
25.3. Cray Peculiarities Carter [188, 1991] describes how a Cray computer at the NASA Ames Laboratories produced puzzling results that were eventually traced to properties of its floating point arithmetic. Carter used Cholesky factorization on a Cray Y-MP to solve a sparse symmetric positive definite system of order 16146 resulting from a finite element model of the National Aerospace Plane. The results obtained for several computers are shown in Table 25.1, where “Displacement” denotes the largest component of the solution and incorrect digits are set in italics and underlined. Since the coefficient matrix has a condition number of order 1011, the errors in the displacements are consistent with the error bounds for Cholesky factorization. Given that both the Cray 2 and the Cray Y-MP lack guard digits, it is not surprising that they give a less accurate answer than the other machines with a similar unit roundoff. What is surprising is that, even though both machines use 64-bit binary arithmetic with a 48-bit mantissa, the result from the Cray Y-MP is much less accurate than that from the Cray 2. The explanation (diagnosed over the telephone to Carter by Kahn, as he scratched on the back of an envelope) lies in the fact that the Cray 2 implementation of subtraction without a guard digit produces more nearly unbiased results (average error zero), while the Cray Y-MP implementation produces biased results, causing fl(x – y) to be too big if x > y > 0. Since the inner loop of the Cholesky algorithm contains an operation of the form a i i = a i i – the errors in the diagonal elements reinforce as the factorization proceeds on the Cray Y-MP, producing a Cholesky factor with a diagonal that is systematically too large. Similarly, since Carter’s matrix has off-diagonal elements that are mostly negative or zero, the Cray Y-MP produces a Cholesky factor with off-diagonal elements that are systematically too large and negative. For the large value of n used by Carter, these two effects in concert are large enough to cause the loss of a further two digits in the answer. An inexpensive way to improve the stability of the computed solution is to
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use iterative refinement in fixed precision (see Chapter 11). This was tried by Carter. He found that after one step of refinement the Cray Y-MP solution was almost as accurate as that from the Cray 2 without refinement.
25.4. Compilers Some of the pitfalls a compiler writer should avoid when implementing floating point arithmetic are discussed by Farnum [364, 1988]. The theme of his paper is that programmers must be able to predict the floating point operations that will be performed when their codes are compiled; this may require, for example, that the compiler writer forgoes certain “optimizations”. In particular, compilers should not change groupings specified by parentheses. For example, the two expressions (1.0E-30 + 1.0E+30) - 1.0E+30 1.0E-30 + (1.0E+30 - 1.0E+30) will produce different answers on many machines. Farnum explains that Compiler texts and classes rarely address the peculiar problems of floating-point computation, and research literature on the topic is generally confined to journals read by numerical analysts, not compiler writers. Many production-quality compilers that are excellent in other respects make basic mistakes in their compilation of floating-point, resulting in programs that produce patently absurd results, or worse, reasonable but inaccurate results.
25.5. Determining Properties of Floating Point Arithmetic Clever algorithms have been devised that attempt to reveal the properties and parameters of a floating point arithmetic. The first algorithms were published by Malcolm [724, 1972] (see Problem 25.3). These have been improved and extended by other authors. Kahan’s paranoia code carries out detailed investigations of a computer’s floating point arithmetic; there are Basic, C, Modula, Pascal, and Fortran versions, all available from netlib. In addition to computing the arithmetic parameters, paranoia tries to determine how well the arithmetic has been implemented (so it can be regarded as a program to test a floating point arithmetic—see the next section). Karpinski [645, 1985] gives an introduction to paranoia for the layman, but the best documentation for the code is the output it produces. Cody has a Fortran routine macher for determining 13 parameters associated with a floating point arithmetic system [222, 1988] (an earlier version was published in the book by Cody and Waite [228, 1980]). Cody notes some
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strange behaviour of compilers and says that he was unable to make his code run correctly on the Cyber 205. A routine based on machar is given in Numerical Recipes [842, 1992, §20.1]. LAPACK contains a routine xLAMCH for determining machine parameters. Because of the difficulty of handling the existing variety of machine arithmetics it is over 850 lines long (including comments and the subprograms it calls). Programs such as machar, paranoia, and xLAMCH are difficult to write; for example, xLAMCH tries to determine the overflow threshold without invoking overflow. The Fortran version of paranoia handles overflow by requiring the user to restart the program, after which checkpointing information previously written to a file is read to determine how to continue. Fortran 90 contains environmental inquiry functions, which for REAL arguments return the precision (PRECISION23), exponent range (RANGE), machine epsilon (EPSILON), largest number (HUGE), and so on, corresponding to that argument [749, 1990]. The values of these parameters are chosen by the implementor to best fit a model of the machine arithmetic due to Brown [150, 1981] (see §25.7.4). Fortran 90 also contains functions for manipulating floating point numbers: for example, to set or return the exponent or fractional part (EXPONENT, FRACTION, SET_EXPONENT) and to return the spacing of the numbers having the same exponent as the argument (SPACING).
25.6. Testing a Floating Point Arithmetic How can we test whether a particular implementation of floating point arithmetic is correct? It is impractical to test a floating point operation with all possible arguments because there are so many of them—about 1019 in IEEE double precision arithmetic, for example! Special techniques are therefore needed that test with a limited number of arguments. A package FPV [784, 1986] from NAG Ltd. attempts to verify experimentally that a floating point arithmetic has been correctly implemented according to its specification. FPV must be supplied by the user with the arithmetic parameters β, t, emin, emax, and with the rounding rule; it then attempts to verify that the arithmetic conforms to these parameters by probing for errors. The tests cover the basic arithmetic operations (including square root) but not the elementary functions. FPV, which is available in both Fortran 77 and Pascal versions, adopts an approach used in an earlier program FPTST [906, 1981], [907, 1984] by Schryer of AT&T Bell Laboratories: it tests arithmetic operations with a limited set of operands that are regarded as being most likely to reveal errors. This approach is based on the premise that errors are most likely to occur as “edge effects”, induced by operands close to a discontinuity or boundary of the floating point number system (such as a power of 23
Intrinsic function names are shown in parentheses.
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the base β). FPV and FPTST have both revealed implementation errors in floating point arithmetics on commercial computers. Errors detected include multiplication and negation producing unnormalized results, x * y differing from (–x) * (–y), and the product of two numbers greater than 1 being less than 1! Wichmann [1078, 1992] suggests that it was probably revelations such as these that led the UK Ministry of Defence to issue an interim standard prohibiting the use of floating point arithmetic in safety-critical systems. A Fortran package ELEFUNT by Cody and Waite contains programs to test the elementary functions [221, 1982], [228, 1980]; the package is available from netlib. It checks identities such as cos(x ) = cos(x/3)(4 cos2(x/3) – 1), taking care to choose arguments x for which the errors in evaluating the identities are negligible. A package CELEFUNT serves the same purpose for the complex elementary functions [224, 1993]. Tang [988, 1990] develops table-driven techniques for testing the functions exp and log.
25.7. Portability Software is portable if it can be made run on different systems with just a few straightforward changes (ideally, we would like to have to make no changes, but this level of portability is often impossible to achieve). Sometimes the word “transportable” is used to describe software that requires certain well-defined changes to run on different machines. For example, Demmel, Dongarra, and Kahan [290, 1992] describe LAPACK as “a transportable way to achieve high efficiency on diverse modern machines”, noting that to achieve high efficiency the BLAS need to be optimized for each machine. A good example of a portable collection of Fortran codes is LINPACK. It contains no machine-dependent constants and uses the PFORT subset of Fortran 77; it uses the level-1 BLAS, so, ideally, optimized BLAS would be used on each machine. 25.7.1. Arithmetic Parameters Differences between floating point arithmetics cause portability problems. First, what is meant by REAL and DOUBLE PRECISION in Fortran varies greatly between machines, as shown by the following table:
Second, for a given level of precision, u, the various arithmetic parameters such as base, unit roundoff, largest and smallest machine numbers, and the
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type of rounding, can all vary. Some possible ways for a code to acquire these parameters are as follows. (1) The parameters are evaluated and embedded in the program in PARAMETER and DATA statements. This is conceptually the simplest approach, but it is not portable. (2) Functions are provided that return the machine parameters. Bell Laboratories’ PORT library [405, 1978] has three functions: R1MACH(k) 1 < k < 5 D1MACH(k) 1 < k < 5 I1MACH(k) 1 < k < 16
real parameters, double precision parameters, integer parameters.
R1MACH returns the underflow and overflow thresholds, the smallest and largest relative spacing (β –t, β 1-t respectively), and log10 β, where β is the base and t the number of digits in the mantissa. I1MACH returns standard input, output and error units and further floating point information, such as β, t, and the minimum and maximum exponents emin and emax. These functions do not carry out any computation; they contain DATA statements with the parameters for most common machines in comment lines, and the relevant statements have to be uncommented for a particular machine. This approach is more sensible for a library than the previous one, because only these three routines have to be edited, instead of every routine in the library. The NAG Library takes a similar approach to PORT. Each of the 18 routines in its X02 chapter returns a different machine constant. For example, XO2AJF returns the unit roundoff and X02ALF returns the largest positive floating point number. These values are determined once and for all when the NAG library is implemented on a particular platform using a routine similar to paranoia and machar, and they are hard coded into the Chapter X02 routines. (3) The information is computed at run-time, using algorithms such as those described in §25.5. 25.7.2. 2 x 2 Problems in LAPACK LAPACK contains 10 or so auxiliary routines for solving linear equation and eigenvalue problems of dimension 2. For example, SLAS2 computes the singular values of a 2 x 2 triangular matrix, SLAEV2 computes the eigenvalues and eigenvectors of a 2 x 2 symmetric matrix, and SLASY2 solves a Sylvester equation AX – XB = C where A and B have order 1 or 2. These routines need to be reliable and robust because they are called repeatedly by higher-level solvers. Because LAPACK makes minimal assumptions about the underlying floating point arithmetic, the 2 x 2 routines are nontrivial to write and are surprisingly long: the counts of executable statements are 39 for SLAS2, 61 for
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SLAEV2, and 207 for SLASY2. If the availability of extended precision arithmetic (possibly simulated using the working precision) or IEEE arithmetic can be assumed, the codes can be simplified significantly. Complicated and less efficient code for these 2 x 2 problem solvers is a price to be paid for portability across a wide range of floating point arithmetics. 25.7.3. Numerical Constants How should numerical constants be specified, e.g., for a Gauss quadrature rule or a Runge–Kutta method? Some compilers limit the number of significant digits that can be specified in a DATA statement or an assignment statement. One possibility is to use rational approximations, but again the number of digits required to cater for all precision may be too large for some compilers. Another scheme is to store a carefully designed table of leading and trailing parts of constants. A constant such as π is best computed via a statement such as pi = 4.0*atan(1.0), if you trust your language’s mathematics library. 25.7.4. Models of Floating Point Arithmetic The model (2.4) of floating point arithmetic contains only one parameter, the unit roundoff. Other more detailed models of floating point arithmetic have been proposed, with the intention of promoting the development of portable mathematical software. A program developed under a model and proved correct for that model has the attraction that it necessarily performs correctly on any machine for which the model is valid. The first detailed model was proposed by van Wijngaarden [1046, 1966]; it contained 32 axioms and was unsuccessful because it was “mathematically intimidating” and did not cover the CDC 600, an important high-performance computer of that time [635, 1991]. A more recent model is that of Brown [150, 1981]. Brown’s model contains four parameters: the base β, the precision t, and the minimum and maximum exponents emin and emax together with a number of axioms describing the behaviour of the floating point arithmetic. Aberrant arithmetics are accommodated in the model by penalizing their parameters (for example, reducing t by 1 from its actual machine value if there is no guard digit). Brown builds a substantial theory on the model and gives an example program to compute the 2-norm of a vector, which is accompanied by a correctness proof for the model. Brown’s model is intended to cater to diverse computer arithmetics. In fact, “any behaviour permitted by the axioms of the model is actually exhibited by at least one commercially important computer” [150, 1981, p. 457]. This broadness is a weakness. It means that there are important properties shared by many (but not all) machines that the model cannot reflect, such as the exactness of fl(x – y) described in Theorem 2.4. As Kahan [630,
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1981] notes, “Programming for the [IEEE] standard is like programming for one of a family of well-known machines, whereas programming for a model is like programming for a horde of obscure and ill-understood machines all at once.” Although Brown’s model was used in the Ada language to provide a detailed specification of floating point arithmetic, the model is still somewhat controversial. Wichmann [1077, 1989] gives a formal specification of floating point arithmetic in the VDM specification language based on a modification of Brown’s model. The most recent model is the Language Independent Arithmetic (LIA1) [702, 1993]. The LIA-1 specifies floating point arithmetic far more tightly than Brown’s model. It, too, is controversial; an explanation of flaws in an earlier version (know then as the Language Compatible Arithmetic Standard) was published by Kahan [635, 1991]. For a more detailed critique of models of floating point arithmetic see Priest [844, 1992].
25.8. Avoiding Underflow and Overflow A classic example showing how care is needed to avoid underflow and overflow is the evaluation of a vector 2-norm, For about half of all machine numbers x, x2 either underflows or overflows! Overflow is avoided by the following algorithm: s =0 for i = l:n s = s + (x(i)/t)2 end The trouble with this algorithm is that it requires n divisions and two passes over the data (the first to evaluate so it is slow. (It also involves more rounding errors than the unscaled evaluation, which could be obviated by scaling by a power of the machine base.) Blue [128, 1978] develops a onepass algorithm that avoids both overflow and underflow and requires between O and n divisions, depending on the data, and he gives an error analysis to show that the algorithm produces an accurate answer. The idea behind the algorithm is simple. The sum of squares is collected in three accumulators, one each for small, medium, and large numbers. After the summation, various logical tests are used to decide how to combine the accumulators to obtain the final answer. The original, portable implementation of the level-l BLAS routine xNRM2 (listed in [307, 1979]) was written by C. L. Lawson in 1978 and, according to
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Lawson, Hanson, Kincaid, and Krogh [694, 1979], makes use of Blue’s ideas. However, xNRM2 is not a direct coding of Blue’s algorithm and is extremely difficult to understand—a classic piece of “Fortrans spaghetti”! Nevertheless, the routine clearly works and is reliable, because it has been very widely used without any reported problems. Lawson’s version of xNRM2 has now been superseded in the LAPACK distribution by a concise and elegant version by S. J. Harnmarling, which implements a one-pass algorithm; see Problem 25.5. A special case of the vector 2-norm is the Pythagorean sum, which occurs in many numerical computations. One way to compute it is by a beautiful cubically convergent, square-root-free iteration devised by Moler and Morrison [773, 1983 ; see Problem 25.6. LAPACK has a routine xLAPY2 that computes and another routine xLAPY3 that computes both routines avoid overflow by using the algorithm listed at the start of this section with n = 2 or 3. Pythagorean sums arise in computing the l-norm of a complex vector:
The level-l BLAS routine xCASUM does not compute the l-norm, but the more easily computed quantity
The reason given by the BLAS developers is that it was assumed that users would expect xCASUM to compute a less expensive measure of size than the 2norm [694, 1979]. This reasoning is sound, but many users have been confused not to receive the true l-norm. See Problem 6.16 for more on this pseudo 1norm. Another example where underflow and overflow can create problems is in complex division. If we use the formula
then overflow will occur whenever cord exceeds the square root of the overflow threshold, even if the quotient (a + ib)/(c + id) does not overflow. Certain Cray and NEC machines implement complex division in this way [290, 1992]; on these machines the exponent range is effectively halved from the point of view of the writer of robust software. Smith [929, 1962] suggests a simple way to avoid overflow: if |c| > |d| use the following formula, obtained by multiplying the numerators and denominators by c–1 , (25.1)
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and if |d| > |c| use the analogous formula involving d–1. Stewart [947, 1985] points out that underflow is still possible in these formulae, and suggests a more complicated algorithm that avoids both underflow and overflow. Demmel [280, 1984] discusses in detail the effects of underflow on numerical software and analyses different ways of implementing underflows, the main two of which are flush to zero and gradual underflow (as used in IEEE standard arithmetic). Cody [222, 1988] makes the following observations: The variety in the treatment of underflow is wondrous. There exist machines in which the exponent wraps around and underflow results in a large floating-point number, perhaps with an accompanying sign change. There are also machines on which underflow is replaced by the smallest in magnitude nonzero number, machines on which small positive numbers vanish when added to themselves, and machines on which such small numbers vanish when multiplied by 1.0. Finally, on IEEE-style machines, underflow may be graceful.
25.9. Multiple Precision Arithmetic If we wish to work at a precision beyond that provided by a Fortran compiler’s DOUBLE PRECISION (or the quadruple precision supported as an extension by some compilers) there are two basic approaches. The first is to implement higher precision arithmetic using arrays of double precision variables, where the unevaluated sum of the elements of an array represents a high precision variable. Arithmetic can be implemented in this system using techniques such as those of Dekker [275, 1971] (cf. compensated summation, described in §4.3), as refined by Linnainmaa [706, 1981] and Priest [843, 1991], [844, 1992]. These techniques require a “well-behaved” floating point arithmetic, and are not easy to apply. The second approach is to use arrays to represent numbers in “standard” floating point form with a large base and a long mantissa spread across the elements of an array. All the software described in this section is of the second type. The first major piece of multiple precision Fortran software was Brent’s package [145, 1978], [146, 1978]. This is a portable collection of Fortran 66 subroutines with wide capabilities, including the evaluation of special functions. Brent’s package represents multiple precision numbers as arrays of integers and operates on them with integer arithmetic. An interface between Brent’s package and the Fortran precompiled Augment [254, 1979] is described by Brent, Hooper, and Yohe [147, 1980]; it makes Brent’s package much easier to use.
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Recently, Bailey has written a multiple precision arithmetic package MPFUN [44, 1993], which consists of about 10,000 lines of Fortran 77 code in 87 subprograms. In MPFUN, a multiprecision (MP) number is a vector of single precision floating point numbers; it represents a number in base 224 (for IEEE arithmetic). Complex multiprecision numbers are also supported. MPFUN routines are available to perform the basic arithmetic operations, evaluate nth roots and transcendental functions, compare MP numbers, produce a random MP number, solve polynomial equations, and perform other operations. For many of these routines both simple and advanced algorithms are provided; the advanced algorithms are intended for computations at precision levels above 1000 digits. One advanced algorithm is a fast Fourier transform technique for evaluating the convolution involved in a multiplication. Another is used for division: x/y is evaluated as x * (1/y ), where 1/y is evaluated by Newton’s method (see Problem 2.26), which involves only multiplications. An interesting aspect of Newton’s method in this context is that it can be implemented with a precision level that doubles at each stage, since the iteration damps out any errors. Another interesting feature of a variable precision environment is that numerical instability can be tackled simply by increasing the precision. As an example, MPFUN does complex multiplication using the 3M method
which uses three real multiplications instead of the usual four. As we saw in §22.2.4, the 3M method produces a computed imaginary part that can have large relative error, but this instability is not a serious obstacle to its use in MPFUN. Bailey provides a translator that takes Fortran 77 source containing directives in comments that specify the precision level and which variables are to be treated as multiprecision, and produces a Fortran 77 program containing the appropriate multiprecision subroutine calls. He also provides a Fortran 90 version of the package that employs derived data types and operator extensions [45, 1994]. This Fortran 90 version is a powerful and easy to use tool for doing high precision numerical computations. Bailey’s own use of the packages includes high precision computation of constants such as π [42, 1988] and Euler’s constant γ. A practical application to a fluid flow problem is described by Bailey, Krasny, and Pelz [46, 1993]. They found that converting an existing code to use MPFUN routines increased execution times by a factor of about 400 at 56 decimal digit precision, and they comment that this magnitude of increase is fairly typical for multiple precision computation. Another recent Fortran 77 program for multiple precision arithmetic is the FM package of Smith [926, 1991]. It is functionally quite similar to Bailey’s MPFUN. No translator is supplied for converting from standard Fortran 77
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source code to code that invokes FM, but Smith notes that the general purpose precompiled Augment [254, 1979] can be used for this purpose. Fortran routines for high precision computation can also be found in Numerical Recipes [842, 1992, §20.6], and high precision numerical computation is supported by many symbolic manipulation packages, including Maple [199, 1991] and Mathematical [1109, 1991]. A Pascal-like programming language called Turing [579, 1988] developed at the University of Toronto in the 1980s is the basis of an extension called Numerical Turing, developed by Hull and his co-workers [591, 1985]. Numerical Turing includes decimal floating point arithmetic whose precision can be dynamically varied throughout the course of a program, a feature argued for in [590, 1982] and implemented in hardware in [231, 1983]. An extension to the level-2 BLAS (see Appendix D) is proposed in [313, 1988, App. B], comprising routines having the same specifications as those in the standard BLAS but which calculate in extended precision.
25.10. Patriot Missile Software Problem A report from the United States General Accounting Office begins “On February 25, 1991, a Patriot missile defense system operating at Dhahran, Saudi Arabia, during Operation Desert Storm failed to track and intercept an incoming Scud. This Scud subsequently hit an Army barracks, killing 28 Americans” [1036, 1992]. The report finds that the failure to track the Scud missile was caused by a precision problem in the software. The computer used to control the Patriot missile is baaed on a 1970s design and uses 24-bit arithmetic. The Patriot system tracks its targets by measuring the time it takes for radar pulses to bounce back from them. Time is recorded by the system clock in tenths of a second, but is stored as an integer. To enable tracking calculations the time is converted to a 24 bit floating point number. Rounding errors in the time conversions cause shifts in the system’s “range gate”, which is used to track the target. On February 11, 1991 the Patriot Project Office received field data identifying a 20% shift in the Patriot system’s range gate after the system had been running continuously for 8 hours. This data implied that after 20 consecutive hours of use the system would fail to track and intercept a Scud. Modified software that compensated for the inaccurate time calculations was released on February 16 by army officials. On February 25, Alpha Battery, which was protecting the Dhahran Air Base, had been in continuous operation for over 100 hours. The inaccurate time calculations caused the range gate to shift so much that the system could not track the incoming Scud. On February 26, the next day, the modified software arrived in Dhahran. Table 25.2, taken from [1036, 1992], shows clearly how, with increasing time of operation, the
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Table 25.2. Effect of extended run time on Patriot missile operation. Hours Seconds Calculated time (seconds) 0 0 0 1 3600 3599.9966 28800 28799.9725 8 20a 72000 71999.9313 48 172800 172799.8352 259200 259199.7528 72 100 b 360000 359999.6567 c a b c
Inaccuracy Approximate shift in range gate (meters) (seconds) 0 0 7 .0034 55 .0275 137 .0687 .1648 330 494 .2472 687 .3433
For continuous operation exceeding 20 hours target is outside range gate. Alpha battery ran continuously for about 100 hours. Corrected value.
Patriot lost track of its target. Note that the numbers in Table 25.2 are consistent with a relative error of 2–20 in the computer’s representation of 0.1, this constant being used to convert from the system clock’s tenths of a second to seconds (2–20 is the relative error introduced by chopping 0.1 to 23 bits after the binary point).
25.11. Notes and References The issues discussed in this chapter are rarely treated in textbooks. One book that is worth consulting is Miller’s Engineering of Numerical Software [759, 1984], which is concerned with the designing of reliable numerical software and presents practical examples. Kahan’s paper [627, 1972] is not, as the title suggests, a survey of error analysis, but treats in detail some specific problems, including solution of a quadratic equation and summation, giving error analysis, programs illustrating foibles of Fortran compilers, and ingenious workarounds. The collection of papers improving Floating-Point Programming [1061, 1990] contains, in addition to an introduction to floating point arithmetic and Brown’s model, descriptions of recent work on embedding interval arithmetic with a super-accurate inner product in Pascal and ADA (see §24.4). There are several algorithms for evaluating a continued fraction; see, for example, Blanch [123, 1964] and Press, Teukolsky, Vetterling, and Flannery [842, 1992, §5.2]. Rounding error analysis for the “bottom up” algorithm is given by Jones and Thron [617, 1974]. A standard set of names and definitions of machine parameters was proposed by the IFIP Working Group on Numerical Software (WG 2.5) [383,
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1978], though these do not appear to have been widely adopted. The use of high precision computations to investigate the Riemann hypothesis and other mathematical problems and conjectures is described by Varga [1052, 1990], who has made use of Brent’s package [145, 1978], [146, 1978].
Problems 25.1. (a) The following MATLAB code attempts to compute µ := min{floating point x : fl(1 + x) > 1}, under the assumption that the base β = 2. function x = mu() x = 1; xp1 = x + l; while xp1 > 1 x = x/2 ; xp1 = x + 1; end x = 2*x; On my workstation, running this code gives >> mu ans = 2.2204e-016 Try this code, or a translation of it into another language, on any machines available to you. Are the answers what you expected? (b) Under what circumstances might the code in (a) fail to compute µ? (Hint: consider optimizing compilers.) (c) What is the value of µ in terms of the arithmetic parameters β and t ? Note: the answer depends subtly on how rounding is performed, and in particular on whether double rounding is in effect; see Higham [550, 1991] and Moler [771, 1990]. On a workstation with an Intel 486DX chip (in which double rounding does occur), the following behaviour is observed in MATLAB: >> format hex; format compact % Hexadecimal format. >> x = 2^(-53) + 2^ (-64) + 2^(-105) ; y = [1+x 1 X] y= 3ff0000000000001
3ff0000000000000
3ca0020000000001
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>> x = 2^(-53) + 2^(-64) + 2^(-106); y
=
[1+x 1 x]
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3ca0020000000000
25.2. Show that Smith’s formulae (25.1) can be derived by applying Gaussian elimination with partial pivoting to the 2 x 2 linear system obtained from (c + id)(x + iy) = a + ib. 25.3. The following MATLAB function implements an algorithm of Malcolm [724, 1972] for determining the floating point arithmetic parameters β and t. function [beta, t] = param(x) % a and b are floating point variables. a = 1; while (a+l) - a == 1 a = 2*a; end b=2; while (a+b) == a b = 2*b ; end beta = - a; t = 1; a = beta; while (a+l) - a == 1 t = t+l; a= a*beta; end Run this code, or a translation into another language, on any machines available to you. Explain how the code works. (Hint: consider the integers that can be represented exactly as floating point numbers.) Under what circumstances might it fail to produce the desired results? 25.4. Hough [585, 1989] formed a 512 x 512 matrix using the following Fortran random number generator: subroutine matgen (a, lda, n, b, norms) REAL a(lda,1), b(1), norma c init = 1325 norms = 0.0 do 30 j = 1,n do 20 i = 1,n
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init = mod(3125*init,65536) a(i, j) = (init - 32768 .0)/16384.0 norms = max(a(i, j), norms) 20 continue 30 continue He then solved a linear system using this matrix, on a workstation that uses IEEE single precision arithmetic. He found that the program took an inordinate amount of time to run and tracked the problem to underflows. On the system in question underflows are trapped and the correct result recomputed very slowly in software. Hough picks up the story: I started printing out the pivots in the program. They started out as normal numbers like 1 or – 10, the suddenly dropped to about 1e-7, then later to 1e-14, and then: k k k k k k k k k k k k k k k k
82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97
pivot pivot pivot pivot pivot pivot pivot pivot pivot pivot pivot pivot pivot pivot pivot pivot
-1.8666e-20 -2.96595e-14 2.46156e-14 2.40541e-14 -4.99053e-14 1.7579e-14 1.69295e-14 -1.56396e-14 1.37869e-14 -3.10221e-14 2.35206e-14 1.32175e-14 -7.77593e-15 1.34815e-14 -1.02589e-21 4.27131e-22
k k k k k k k k k k k k k k k k k
98 pivot 1.22101e-21 99 pivot -7.12407e-22 100 pivot -1.75579e-21 101 pivot 3.13343e-21 102 pivot -6.99946e-22 103 pivot 3.82048e-22 104 pivot 8.05538e-22 105 pivot -1.18164e-21 106 pivot -6.349e-22 107 pivot -2.48245e-21 108 pivot -8.89452e-22 109 pivot -8.23235e-22 110 pivot 4.40549e-21 111 pivot 1.12387e-21 112 pivot -4.78853e-22 113 pivot 4.38739e-22 114 pivot 7.3868e-28
SIGFPE 8: numerical exception, CHK, or TRAP stopped at daxpy+0x18c: mov1 a4@(0xe10),a3@ Explain this behaviour. 25.5. The version of the level-l BLAS routine xNRM2 that is distributed with LAPACK implements the following algorithm for computing ||x||2 , t=0 s=1 for i = l:n
PROBLEMS
511
if |xi | > t s = 1 + s (t/xi )2 t = |xi | else s = s + (xi /t)2 end end
Prove that the algorithm works and describe its properties. 25.6. (Moler and Morrison [773, 1983], Dubrulle [323, 1983]) This MATLAB M-file computes the Pythagorean sum using an algorithm of Moler and Morrison. function p = pythag(a, b) %PYTHAG Pythagorean sum. P = max(abs(a) , abs(b)) ; q = min(abs(a) , abs(b)) ; while 1 r = (q/p)^2; if r+4 == 4, return, end s = r/(4+ r); p = p+2*s*p; q = s*q; fprintf(’p = %19.15e, q = %19.15e\n’, p, q) end The algorithmic immune to underflow and overflow (unless the result overflows), is very accurate, and converges in at most three iterations, assuming the unit roundoff u > 10–20. Example: p = pythag(3,4); (p-5)/5 p = 4.986301369863014e+000, q = 3.698630136986301e-001 p = 4.999999974188253e+000, q = 5.080526329415358e-004 p = 5.000000000000001e+000, q = l.311372652397091e-012 ans = 1.7764e-016 The purpose of this problem is to show that pythag is Halley’s method applied to a certain equation. Halley’s method for solving f(x) = 0 is the iteration
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IN
FLOATING P OINT A RITHMETIC
where fn, f´n and f´´n denote the values of f and its first two derivatives at xn. (a) For given x 0 and y 0 such that 0 < y 0 < x 0 , the Pythagorean sum is a root of the equation f(x) = x 2 – p 2 = 0. Show that Halley’s method applied to this equation gives
Show that if 0 < x0. < p = then xn < xn+1 < p for all n. Deduce that y n := is defined and that
Confirm that pythag implements a scaled version of these equations. (b) Show that
which displays the cubic convergence of the iteration. 25.7. (Incertis [604, 1985]) (a) Given a skew-symmetric matrix Y with ||Y||2 < 1, show that there is a real, symmetric X satisfying X2 = I + Y2 such that X + Y is orthogonal. (b) Consider the following iteration, for P0 , Q0 with ||P0||2 > ||Q0||2, which generalizes to matrices the "pythag" iteration in Problem 25.6. for k = R k = (Q k P k - 1 ) 2 S k = R k(4I + R k) - 1 Pk+1 = Pk + 2S k P k Qk+l = SkQk end Show that this iteration can be used to compute X in part (a) (c) Investigate the convergence of the iteration in (b) for general P0 and Q0. 25.8. Why might we prefer the expression intended to be robust in floating point arithmetic?
in an algorithm
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Chapter 26
A Gallery of Test Matrices
Many tricks or treats associated with the Hilbert matrix may seem rather frightening or fascinating. — MAN-DUEN CHOI, Tricks or Treats with the Hilbert Matrix (1983)
I start by looking at a 2 by 2 matrix. Sometimes I look at a 4 by 4 matrix. That’s when things get out of control and too hard. Usually 2 by 2 or 3 by 3 is enough, and I look at them, and I compute with them, and I try to guess the facts. First, think of a question. Second, I look at examples, and then third, guess the facts. — PAUL R. HALMOS 2 4 (1991)
When people look down on matrices, remind them of great mathematicians such as Frobenius, Schur, C. L. Siegel, Ostrowski, Motzkin, Kac, etc., who made important contributions to the subject. — OLGA TAUSSKY, How I Became a Torchbearer for Matrix Theory (1988)
24
From interviews by Albers in [9, 1991].
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Ever since the first computer programs for matrix computations were written in the 1940s, researchers have been devising matrices suitable for test purposes and investigating the properties of these matrices. In the 1950s and 1960s it was common for a whole paper to be devoted to a particular test matrix: typically its inverse or eigenvalues would be obtained in closed form. Early collections of test matrices include those of Newman and Todd [796, 1958] and Rutishauser [886, 1968]; most of Rutishauser’s matrices come from continued fractions or moment problems. Two well-known books present collections of test matrices. Gregory and Karney [482, 1969] deal exclusively with the topic, while Westlake [1076, 1968] gives an appendix of test matrices. In this chapter we present a gallery of matrices. We describe their properties and explain why they are useful (or not so useful, as the case may be) for test purposes. The coverage is limited. A comprehensive, up-to-date, and well-documented collection of parametrized test matrices may be found in the Test Matrix Toolbox, described in Appendix E. Of course, MATLAB itself contains a number of special matrices that can be used for test purposes (type help specmat). Several other types of matrices would have been included in this chapter had they not been discussed elsewhere in the book. These include magic squares (Problem 6.4), the Kahan matrix (8.10), Hadamard matrices (§9.3), and Vandermonde matrices (Chapter 21). The matrices described here can be modified in various ways while still preserving some or all of their interesting properties. Among the many ways of constructing new test matrices from old are l
Similarity transformations A
l
Unitary transformations A
l
l
Kronecker products A routine kron) . Powers A
A
X – 1 AX. U AV, where U*U = V*V = I. B or B
A (for which MATLAB has a
Ak.
26.1. The Hilbert and Cauchy Matrices The Hilbert matrix H n with elements hij = 1/(i + j – 1), is perhaps the most famous of all test matrices. It was widely used in the 1950s and 1960s for testing inversion algorithms and linear equation solvers. Its attractions were threefold: it is very ill conditioned for even moderate values of n, formulae are known for the elements of the inverse, and the matrix arises in a practical problem: least squares fitting by a polynomial expressed in the monomial basis.
26.1 T HE H ILBERT
AND
C AUCHY M ATRICES
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Despite its past popularity and notoriety, the Hilbert matrix is not a good test matrix. It is too special. Not only is it symmetric positive definite, but it is totally positive. This means, for example, that Gaussian elimination without pivoting is guaranteed to produce a small componentwise relative backward error (as is Cholesky factorization). Thus the Hilbert matrix is not a typical ill-conditioned matrix. The (i, j) element of the inverse of the Hilbert matrix Hn is the integer
(26.1) and (26.2) There are many ways to rewrite the formula (26.1). These formulae are best obtained as special cases of those for the Cauchy matrix below. The Cholesky factor Rn of the inverse of the Hilbert matrix is known explicitly, as is Rn-1: (26.3) (26.4) One interesting application of these formulae is to compute the eigenvalues of H n as the squares of the singular values of Rn ; if Rn is evaluated from (26.3) and the one-sided Jacobi algorithm is used to compute the singular values then high relative accuracy is obtained for every eigenvalue, as shown by Mathias [732, 1995]. The condition number of the Hilbert matrix grows at an exponential rate: (H k2 n) ~ exp(3.5n) [1004, 1954]. See Table 26.1 for the first few condition numbers (these were obtained by computing the inverse exactly using MATLAB’S Symbolic Math Toolbox [204, 1993] and then computing the norm of the numeric representation of the inverse; the numbers given are correct to the figures shown). It is an interesting fact that the matrix = ( 1 / (i + j) ) (a submatrix of H n+l ) satisfies µ n := = π + O(1/log n) as as proved by Taussky [996, 1949]. The convergence to π is very slow: µ 200 = 2.01, µ300 = 2.08, µ400 = 2.12. That H n -1 is known explicitly is not as useful a property for testing an inversion algorithm as it might appear, because Hn cannot be stored exactly in floating point arithmetic. This means that when we attempt to invert H n we at best invert H n + ∆H (the matrix actually stored on the computer),
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Table 26.1. Condition numbers of Hilbert and Pascal matrices. n 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
K2(Hn) 1.9281e1 5.2406e2 1.5514e4 4.7661e5 1.4951e7 4.7537e8 1.5258e10 4.9315e11 1.6026e13 5.2307e14 1.7132e16 5.6279e17 1.8534e19 6.l166e20 2.0223e22
K2(Pn) 6.8541e0 6.1984el 6.9194e2 8.5175e3 1.1079e5 1.4934e6 2.0645e7 2.9078e8 4.1552e9 6.0064e10 8.7639e11 1.2888e13 1.9076e14 2.8396e15 4.2476e16
w h e r e | ∆H| < uH n , and (Hn + ∆H)–1 can differ greatly from Hn-1, in view of the ill conditioning. A possible way round this difficulty is to start with the integer matrix H n -1 , but its entries are so large that they are exactly representable in IEEE double precision arithmetic only for n less than 13. The Hilbert matrix is a special case of a Cauchy matrix C n whose elements are cij = 1/(xi + yj), where x, y are given n-vectors (take xi = yi = i – 0.5 for the Hilbert matrix). The following formulae give the inverse and determinant of Cn, and therefore generalize those for the Hilbert matrix. The (i, j) element of Cn-1 is
and
the latter formula having been published by Cauchy in 1841 [189, 1841, pp. 151–159]. The LDU factors of Cn have been found explicitly by Gohberg
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517
and Koltracht [454, 1990]: lkk = ukk = 1 and
It is known that Cn is totally positive if 0 < x1 < · · · < xn and 0 < yl < · · · < yn (as is true for the Hilbert matrix) [998, 1962, p. 295]. Interestingly, the sum of all the elements of [667, 1973, Ex. 44, §1.2.3].
26.2. Random Matrices Random matrices are widely used for test purposes. Perhaps the earliest use of random matrices in numerical analysis was by von Neumann and Goldstine [1057, 1947], [462, 1951], who estimated the size of their error bounds for matrix inversion (see §9.6) for the case of random matrices; to do this, they had to estimate the condition number of a random matrix. Intuitively, one might expect random matrices to be good at revealing programming errors and unusual behaviour of algorithms, but this expectation is not necessarily correct. For example, Miller [759, 1984, pp. 96-97] describes a mutation experiment involving Fortran codes for Gaussian elimination without pivoting, Gaussian elimination with partial pivoting, and Gauss– Jordan elimination with partial pivoting. For each code, all possible mutants were generated, where a mutant is obtained by making a single typographical change to the source code. All the mutants were tested on a single random linear system Ax = b, with known solution, where a ij was chosen from the uniform [– 1, 1] distribution. Many mutants were detected by their failure to pass the test of producing a solution with forward error less than a tolerance. However, some mutants passed this test, including all those that solve a system correctly in exact arithmetic; mutants in the latter class include those that select an incorrect pivot row and thus implement a numerically unstable algorithm. A conclusion to be drawn from Miller’s experiment is that random test data can reveal some programming errors, but will not reveal all. A good example of a problem for which random test matrices are very poor at revealing algorithmic weaknesses is matrix condition number estimation. The popular condition estimation algorithms can yield poor estimates but, in
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practice, never produce them for a random matrix (see Chapter 14). The role of random matrices here is to indicate the average quality of the estimates. Edelman [340, 1993] summarizes the properties of random matrices well when he says that What is a mistake is to psychologically link a random matrix with the intuitive notion of a “typical” matrix or the vague concept of “any old matrix.” In contrast, we argue that “random matrices” are very special matrices. The larger the size of the matrices the more predictable they are because of the central limit theorem. Various results are known about the behaviour of matrices with elements from the normal N(0, 1) distribution. Matrices of this type are generated by MATLAB’S randn function. Let An denote an n x n matrix from this distribution and let E(·) be the expectation operator. Then, in the appropriate probabilistic sense, the following results hold as n (real data),
(26.5)
(complex data),
(26.6)
(real data), (complex data), (real or complex data).
(26.7) (26.8) (26.9)
For details of (26.5)-(26.8) see Edelman [335, 1988]. Edelman conjectures that the condition number results are true for any distribution with mean O—in particular, the uniform [– 1, 1] distribution used by MATLAB’S rand function. The results (26.5) and (26.6) show that random matrices from the normal N(0, 1) distribution tend to be very well conditioned. The spectral radius result (26.9) has been proved as an inequality by Geman [432, 1986] for independent identically distributed random variables aij with zero mean and unit variance, and computer experiments suggest the approximate equality for the standard normal distribution [432, 1986]. A question of interest in eigenvalue applications is how many eigenvalues of a random real matrix are real. The answer has been given by Edelman, Kostlan, and Shub [344, 1994]: denoting by En the expected number of real eigenvalues of an n x n matrix from the normal N(0, 1) distribution,
Thus the proportion of real eigenvalues, En/n, tends to zero as n formulae for En for finite n are also given in [344, 1994].
Exact
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26.3. “Randsvd” Matrices By randsvd25 we mean a matrix A formed as A = UΣVT, where are random orthogonal matrices and Σ = diag(σ i ) is a given matrix of singular values. This type of matrix has a predetermined singular value distribution (and 2-norm condition number), but is, nevertheless, random. Randsvd matrices have been widely used as test matrices, for example for condition estimators [534, 1987], [946, 1980], and in the LAPACK testing software. Singular value distributions of interest include, with α := K2(A) > 1 a parameter, 1. one large singular value: σ1 = 1, σi = α–1, i = 2:n; 2. one small singular value: σi = 1, i = 1:n – 1, σ n = α– l ; 3. geometrically distributed singular values: σi = β 1–i,
i
=
1:n, where
4. arithmetically distributed singular values: σ i = 1 – (1 – α - l )(i – 1)/(n – 1), i = 1:n. To be precise about what we mean by “random orthogonal matrix” we specify matrices from the Haar distribution, which is a natural distribution over the space of orthogonal matrices, defined in terms of a measure called the Haar measure [780, 1982, 52.1.4]. If A has elements from the normal N(0, σ 2) distribution and A has the QR factorization A = QR, with the factorization normalized so that the diagonal elements of R are positive, then Q is from the Haar distribution, for any variance σ 2 [100, 1979], [946, 1980]. If we compute Q from this prescription, the cost is 2n3 flops. For our randsvd application, a more efficient approach is based on the following result of Stewart [946, 1980]. Theorem 26.1 (Stewart). Let the independent vectors x i have elements from the normal N(0, 1) distribution for i = 1:n – 1. Let Pi = diag where is the Householder transformation that reduces xi to rii el. Then the product Q = DP1P2 . . . Pn-l is a random orthogonal matrix from the Haar distribution, where D = diag(sign(rii ) ) . This result allows us to compute a product form representation of a random n x n orthogonal matrix from the Haar distribution in O(n2 ) flops. If we implicitly compute U and using the construction in Theorem 26.1, and then form A = UΣVT, exploiting the structure, the cost 25
Randsvd is the name of the MATLAB M-file in the Test Matrix Toolbox that generates matrices of this type.
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is m3 + n3 flops. The alternative, which involves obtaining U and V from the QR factorization of random matrices A, is about twice as expensive. Note that forming UΣVT with U and V single Householder matrices, as is sometimes done in the literature, is not recommended, as it produces matrices of a very special form: diagonal plus a rank-2 correction. The construction of randsvd matrices is, of course, easily adapted to produce random symmetric matrices A = QΛQT with given eigenvalues.
26.4. The Pascal Matrix The numbers in Pascal’s triangle satisfy, practically speaking, infinitely many identities. — RONALD L. GRAHAM, DONALD E. KNUTH, and OREN PATASHNIK, Concrete Mathematics (1989) A particularly entertaining test matrix is the Pascal matrix Pn defined by
The rows of Pascal’s triangle appear as anti-diagonals of Pn26: >> p = pascal(6) P= 1 1 1 1 1 1
1 2 3 4 5 6
1 3 6 10 15 21
1 4 10 20 35 56
1 5 15 35 70 126
1 6 21 56 126 252
The earliest references to the Pascal matrix appear to be in 1958 by Newman and Todd [796, 1958] and by Rutishauser [887, 1958] (see also Newman [795, 1962, pp. 240-24 1]); Newman and Todd say that the matrix was introduced to them by Rutishauser. The matrix was independently suggested as a test matrix by Caffney [177, 1963]. Rutishauser [886, 1968, §8] notes that Pn belongs to the class of moment matrices M whose elements are contour integrals
26
In the MATLAB displays below we use the pascal function from the Test Matrix Toolbox. This differs from the pascal function supplied with Matlab 4.2 only in that pascal (n, 2) is rearranged.
521
26.4 T HE P ASCAL M ATRIX
All moment matrices corresponding to a positive weight function w(z) on a contour C are Hermitian positive definite (as is easily verified by considering the quadratic form y*My). The choice C = [0, 1], with weight function w(z) = 1, yields the Hilbert matrix. The Pascal matrix is obtained for C the circle {z : |z – 1| = 1 } and w(z) = (2 πi(z – 1))-1 (not w(z) = (2π)-1 as stated in [886, 1968]); the change of variable z = 1 + exp(iθ ) yields a moment integral with a positive weight function. Remarkably, the Cholesky factor of the Pascal matrix again contains the rows of Pascal’s triangle, now arranged columnwise: >> R = chol(P) R= 1 0 0 0 0 0
1 1 0 0 0 0
1 2 1 0 0 0
1 3 3 1 0 0
1 4 6 4 1 0
1 5 10 10 5 1
The scaled and transposed Cholesky factor S = RT diag(1, –1,1, –1,. . . . (–1)n + 1 ) is returned by pascal (n, 1 ): >> S = pascal(6, 1) S = 1 1 1 1 1 1
0 -1 -2 -3 -4 -5
0 0 1 3 6 10
0 0 0 -1 -4 -10
0 0 0 0 1 5
0 0 0 0 0 -1
It is involuntary: S 2 = I. This special property leads us to several more properties of P = P n . First, since P = SS T , P –1 = S -T S -l = S T S, and so P-1 has integer entries (as is also clear from the fact that det(P) = det(R)2 = 1). Moreover, P = SST = S(STS)S-1 = SP-1S-1, so P and P–1 are similar and hence have the same eigenvalues. In other words, the eigenvalues appear in reciprocal pairs. In fact, the characteristic polynomial π n has a palindromic coefficient vector, which implies the reciprocal eigenvalues property, since π n (λ) = This is illustrated as follows (making use of MATLAB’S Symbolic Math Toolbox):
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>> charpoly(P) ans =
1-351*x+6084*x^2-13869*x^3+6084*x^4-351*x^5+x^6 >> eig(P) ans = 0.0030 0.0643 0.4893 2.0436 15.5535 332.8463
Since P is symmetric, its eigenvalues are its singular values and so we also have that ||P||2 = ||P-1 ||2 and ||P||F = ||P-1 ||F. Now
where for the last equality we used a binomial coefficient summation identity from [477, 1989, p. 161]. Hence, using Stirling’s approximation (n ! ~
Thus Pn is exponentially ill conditioned as n It is worth pointing out that it is not hard to generate symmetric positive definite matrices with determinant 1 and the reciprocal root property. Let X = ZDZ -l where Z is nonsingular and D = diag(±1) ±I. Then X2 = I and the matrix A = X T X has the desired properties. If we choose Z lower triangular then X is the Cholesky factor of A up to a column scaling by diag(±1). The inverse of the Pascal matrix was found explicitly by Cohen [230, 1975]: the (i,j) element is
The Pascal matrix can be made singular simply by subtracting 1 from the (n,n) element. To see this, note that
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This perturbation, ∆P = –enenT, is far from being the smallest one that makes P singular, which is ∆Popt = where λn is the smallest eigenvalue of P and vn is the corresponding unit eigenvector, for is of order (n!)2 /(2 n)! ~ as we saw above. A more subtle property of the Pascal matrix is that it is totally positive. Karlin [644, 1968, p. 137] shows that the matrix with elements ( i, j = 0,1, . . .) is totally positive; the Pascal matrix is a submatrix of this one and hence is also totally positive. From the total positivity it follows that the Pascal matrix has distinct eigenvalues, which (as we already know from the positive definiteness) are real and positive, and that its ith eigenvector (corresponding to the ith largest eigenvalue) has exactly i – 1 sign changes [414, 1959, Thin. 13, p. 105]. T = pascal (n, 2) is obtained by rotating S clockwise through 90 degrees and multiplying by — 1 if n is even: >> T = pascal(6, 2) T= -1 5 -10 10 -5 1
-1 4 -6 4 -1 0
-1 3 -3 1 0 0
-1 2 -1 0 0 0
-1 1 0 0 0 0
-1 0 0 0 0 0
It has the surprising property that it is a cube root of the identity, a property noted by Turnbull [1028, 1929, p. 332]: >> T*T ans = o 0 0 0 0 -1
0 0 0 0 1 -1
0 0 0 -1 2 -1
0 0 1 -3 3 -1
0 -1 4 -6 4 -1
1 -5 10 -lo 5 -1
0 1 0
0 0 1
0 0 0
0 0 0
0 0 0
>> T*T*T ans = 1 0 0
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Figure 26.1. spy(rem(pascal(32),2)). 0 0 0
0 0 0
0 0 0
1 0 0
0 1 0
0 0 1
Finally, we note that it is trivial to plot an approximation to the Sierpinski gasket [824, 1992, §2.2], [478,1992] in MATLAB: simply type spy(rem(pascal (n), 2)). See Figure 26.1. The picture produced by this command is incorrect for large n, however, because the elements of pascal(n) become too large to be exactly representable in floating point arithmetic.
26.5. Tridiagonal Toeplitz Matrices A tridiagonal Toeplitz matrix has the form
Such matrices arise, for example, when discretizing partial differential equations or boundary value problems for ordinary differential equations. The
26.6 C OMPANION M ATRICES
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eigenvalues are known explicitly [884, 1947], [885, 1952], [1006, 1977, pp. 155– 156] : d + 2(ce)1/2 cos(kπ/(n + 1)), k = 1:n. The eigenvalues are also known for certain variations of the symmetric matrix T n ( c, d, c) in which the (1,1) and (n, n) elements are modified; see Gregory and Karney [482, 1969]. The matrix Tn (–1, 2, –1) is minus the well-known second difference matrix, which arises in applying central differences to a second derivative operator. Its inverse has (i, j) element –i(n – j + 1)/(n + 1) (cf. Theorem 14.9). The condition number satisfies k2 (Tn ) ~ (4/π 2 )n 2 . One interesting property of Tn(c, d, e) is that the diagonals of its LU factorization converge as n when Tn is symmetric and diagonally dominant, and this allows some savings in the computation of the LU factorization, as shown by Malcolm and Palmer [725, 1974]. Similar properties hold for cyclic reduction; see Bondeli and Gander [134, 1994].
26.6. Companion Matrices The companion matrix associated with the characteristic polynomial
of A
is the matrix
The Test Matrix Toolbox function compan computes C, via the call C = compan (A), It is easy to check that C has the same characteristic polynomial as A, and that if λ is an eigenvalue of C then is has rank at least n – 1 for any a corresponding eigenvector. Since C λ, C is nonderogatory, that is, in the Jordan form no eigenvalue appears in more than one Jordan block. It follows that A is similar to C only if A is nonderogatory. There are no explicit formulae for the eigenvalues of C, but, perhaps surprisingly, the singular values have simple representations, as found by Kenney and Laub [651, 1988] (see also Kittaneh [660, 1995]):
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Figure 26.2. Pseudospectra of compan(A).
where The compan function is a useful means for generating new test matrices , compan(A) is a nonnormal matrix with the same from old. For any A eigenvalues as A (to be precise, compan(A) is normal if and only if a0 = 1 and ai = 0 for i > 0). Companion matrices tend to have interesting pseudospectra, as illustrated in Figure 26.2. For more information on the pseudospectra of companion matrices see Toh and Trefethen [1007, 1994].
26.7. Notes and References The earliest reference to the Hilbert matrix appears to be [570, 1894], wherein Hilbert obtains the formula (26.2) for det(H n ). The formula (26.1) for Hn-l is taken from Choi [207, 1983], who describes various interesting properties of the Hilbert matrix and its infinite analogue. An excellent reference for the derivation of formulae for the inverse of the Cauchy and Hilbert matrices is Knuth [667, 1973, pp. 35–37]. Another reference for the Cauchy matrix is Tyrtyshnikov [1034, 1991]. The formulae (26.3) and (26.4) for the Cholesky factor and its inverse are from Choi [207, 1983] and Todd [1004, 1954]. Forsythe and Moler [396, 1967] have a chapter devoted to the Hilbert matrix, in which they describe the underlying least squares problem and discuss numerical computation of the inverse. There have been many papers on the
PROBLEMS
527
Hilbert matrix; two of particular interest are by Todd [1004, 1954], [1005, 1961]. Other references on the eigenvalues and condition numbers of random matrices include Edelman [336, 1991], [339, 1992] and Kostlan [671, 1992]. Anderson, Olkin, and Underhill [20, 1987] suggest another way to construct random orthogonal matrices from the Haar distribution, based on products of random Givens rotations. Marsaglia and Olkin [728, 1984] discuss the generation of random correlation matrices (symmetric positive semidefinite matrices with ones on the diagonal). The involuntary triangular matrix pascal (n, 1) arises in the step-size changing mechanism in an ordinary differential equation code based on backward differentiation formulae; see Shampine and Reichelt [912, 1995]. We mention some other collections of test matrices. The Harwell-Boeing collection of sparse matrices, largely drawn from practical problems, is presented by Duff, Grimes, and Lewis [327, 1989], [328, 1992]. Bai [36, 1994] is building a collection of test matrices for large-scale nonsymmetric eigenvalue problems. Zielke [1128, 1986] gives various parametrized rectangular matrices of fixed dimension with known generalized inverses. Demmel and McKenney [299, 1989] present a suite of Fortran 77 codes for generating random square and rectangular matrices with prescribed singular values, eigenvalues, band structure, and other properties. This suite was the inspiration for the randsvd routine in the Test Matrix Toolbox and is part of the testing code for LAPACK (see below). 26.7.1. LAPACK The LAPACK distribution contains a suite of routines for generating test matrices, located in the directory LAPACK/TESTING/MATGEN (in Unix notation). These routines (whose names are of the form xLAxxx) are used for testing when LAPACK is installed and are not described in the LAPACK Users’ Guide [17, 1995].
Problems 26.1. Investigate the spectral and pseudospectral properties of pentadiagnal Toeplitz matrices. See Figure 26.3 (pentoep is a function in the Test Matrix Toolbox that generates matrices of this type). References are Beam and Warming [85, 1993] and Reichel and Trefethen [866, 1992]. 26.2. (R ESEARCH PROBLEM ) Two methods for generating a real orthogonal matrix from the Haar distribution are Stewart’s method (Theorem 26.1), based on Householder transformations, and a method of Anderson, Olkin, and Underhill [20, 1987], based on Givens rotations. Compare the efficiency
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pentoep(32,0,1/2,0,0,1)
inv(pentoep(32,0,1,1,0,.25))
pentoep(32,0,1/2,1,1,1)
pentoep(32,0,1,0,0,1/4)
Figure 26.3. Pseudospectm of 32 x 32 pentadiagonal Toeplitz matrices. of these two methods when used to generate randsvd matrices. Investigate the use of the Givens rotations approach to construct random banded matrices with given singular values and random symmetric banded matrices with given eigenvalues, and compare with the technique of generating a full matrix and then using band reduction (as implemented in the routine randsvd in the Test Matrix Toolbox). 26.3. (RESEARCH PROBLEM) Develop an efficient algorithm for computing a unit upper triangular n x n matrix with prescribed singular values σ1, . . . . σ n , where
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Appendix A
Solutions to Problems we have
1.1 Since
Hence if < 0.01, say, then there is no difference between Erel and practical purposes.
for
1.2. Nothing can be concluded about the last digit before the decimal point. Evaluating y to higher precision yields t 35 40
y 262537412640768743.99999999999925007 262537412640768743.9999999999992500725972
This shows that the required digit is, in fact, 3. The interesting fact that y is so close to an integer was pointed out by Lehmer [697, 1943], who explains its connection with number theory. It is known that y is irrational [959, 1991]. 1.3. 1. 2. 2sin 3. ( x – y)(x + y). Cancellation has not been avoided, but it is now harmless if x and y are known exactly (see also Problem 3.9). 4. sin x/(1 + cos x). 5. c = ((a – b)2 + ab(2sinθ/ 2 ) 2 ) 1 / 2 . 1.4. a + ib = (x + iy)2 = x2 – y2 + 2ixy, so b = 2xy and a = x2 – y2, giving x2 – b2 /(4x 2) = a, or 4x4 – 4ax2 – b2 = 0. Hence
If a > 0 we use this formula with the plus sign, since x2 > 0. If a < 0 this formula is potentially unstable, so we use the rewritten form
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In either case we get two values for x and recover y from y = b /(2x ). Note that there are other issues to consider here, such as scaling to avoid overflow. 1.5. We need a way to compute f(x) = log(1 + x) accurately for all x > 0. A straightforward evaluation of log(1 + x) is not sufficient, since the addition 1 + x loses significant digits of x when x is small. The following method is effective (for another approach, see Hull, Fairgrieve, and Tang [592, 1994]): calculate w = 1 + x and then compute
The explanation of why this method works is similar to that for the method in §1.14.1. We have = (1 + x)(1 + δ), |δ| < u, and if = 1 then |x| < u + u2+ u3+ · · ·, 2 so from the Taylor series f(x) = x(1 – x/2 + x /3 – · · ·) we see that f(x) = x is a correctly rounded result. If then
Defining
=: 1 + z,
so if
(thus x
0) then
Thus, with 1 + θ :=
showing that
is an accurate approximation to f(x) = log(1 + x).
1.7. From (1.4) we have
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This yields the following inequality, which is attainable to first order:
Dividing by The result for
and taking the limit as follows from
gives the expression for
where the inequality is attainable (Cauchy–Schwarz), together with the relation That is easily verified. 1.8. The general solution to the recurrence is
where a, b, and c are arbitrary constants. The particular starting values chosen yield a = 0, b = c = 1, so that
Rounding errors in the evaluation (and in the representation of x1 on a binary machine) cause a nonzero “a” term to be introduced, and the computed values are approximately of the form
for a constant η of order the unit roundoff. Hence the computed iterates rapidly converge to 100. Note that resorting to higher precision merely delays the inevitable convergence to 100. Priest [844, 1992, pp. 54–56] gives some interesting observations on the stability of the evaluation of the recurrence. 1.9. Writing C:= adj(A) = we have
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so – (1/d) A ∆Cb, so |r| < γ3 |A||A-l||b|, which implies the normwise Also, r = b – residual bound. Note that the residual bound shows that will be small if x is a large-normed solution 1.10. Let
Then, since
For any standard summation method we have (see §4.2)
satisfies
Hence, defining
that is,
where
2.1. There are 1 + 2(e max – emin + 1)(β – 1)β t-1 normalized numbers (the “1” is for zero), and 2(β t–1 – 1) subnormal numbers. For IEEE arithmetic we therefore have
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single precision double precision
Normalized Subnormal 1.7 x 107 4.3 x 109 19 1.8 x 10 9 x 1015
2.2. Without loss of generality suppose x > 0. We can write x = m x β e-t, where β t-l < m < β t. The next larger floating point number is x + ∆x, where ∆x = β e - t , and The same upper bound clearly holds for the “next smaller” case, and the lower bound in this case is also easy to see. 2.3. The answer is the same for all adjacent nonzero pairs of single precision numbers. Suppose the numbers are 1 and 1 + (single) = 1 + 2-23. The spacing of the -52 double precision numbers on [1,2] is 2 , so the answer is 229 – 1 5.4 x 108. 2.4. Inspecting the proof of Theorem 2.2 we see that |yi | > β e-1, i = 1, 2, and so we also have |fl(x) – x|/|fl(x)| < u, that is, fl(x) = x /(1 + δ), |δ| < u. Note that this is not the same δ as in Theorem 2.2, although it has the same bound, and unlike in Theorem 2.2 there can be equality in this bound for δ. 2.5. The first part is trivial. Since, in binary notation, x = 0.1100 1100 1100 1100 1100 1100 | 1100 . . . x 2-3, we have, rounding to 24 bits, 0.1100 1100 1100 1100 1100 1101 x 2-3. Thus
and so 2.6. Since the double precision mantissa contains 53 bits, p = 253 = 9007199254740992 9.01 x 1015. For single precision, p = 224 = 16777216 1.68 x 107. 2.7. 1. True, since in IEEE arithmetic fl( a op b) is defined to be the rounded value of a op b, round(a op b), which is the same as round(b op a). 2. True for round to nearest (the default) and round to zero, but false for round to 3. True, because fl(a + a) := round(a + a) = round(2 * a) =: fl(2 * a). 4. True: similar to 3. 5. False, in general.
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6. True for binary arithmetic. Since the division by 2 is exact, the inequality is equivalent to 2a < fl(a + b) < 2b. But a + b < 2b, so, by the monotonicity of rounding, fl(a + b) = round(a + b) < round(2b) = 2b. The lower bound is verified similarly. 2.8. Examples in 3-digit decimal arithmetic are f1((5.01 + 5.02)/2) = f1(10.0/2) = 5.0 and fl((5.02 + 5.04)/2) = f1(10.1/2) = 5.05. The left-hand inequality is immediate, since fl((b – a)/2) > 0. The right-hand inequality can possibly be violated only if a and b are close (otherwise (a + b)/2 is safely less than b, so the inequality will hold). But then, by Theorem 2.5, fl( b – a) is obtained exactly, and the result will have several zero low-order digits in the mantissa because of cancellation. Consequently, if the base is even the division is done exactly, and the right-hand inequality follows by the monotonicity of rounding. Alternatively, and even more simply, for any base we can argue that fl((b – a)/2) < b – a, so that a + fl((b – a)/2) < b and, again, the result follows by the monotonicity of rounding. 2.9.
(binary). Rounded directly to 53 bits this yields 1 – 2-53. But rounded first to 64 bits it yields
and when this number is rounded to 53 bits using the round to even rule it yields 1.0. 2.12. The spacing between the floating point numbers in the interval (1/2,1] is (cf. Lemma 2.1), so |1/x – fl(1/x)| < which implies that |1 – xfl(1/x)| < Thus Since the spacing of the floating point numbers just to the right of 1 is must round to either 1 – or 1.
xfl(1/x)
2.13. If there is no double rounding the answer is 257,736,490. For a proof that combines mathematical analysis with computer searching, see Edelman [343, 1994]. 2.15. The IEEE standard does not define the results of exponentiation. The choice then 0° = 1 can be justified in several ways. For example, if p(x) = p (0) = a0 = a0 x 00, and the binomial expansion (x + y) n = yields 1 = 00 for x = 0, y = 1. For more detailed discussions see Goldberg [457, 1991, p. 32] and Knuth [669, 1992, pp. 406–408].
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2.17. For IEEE arithmetic the answer is no. Since fl( x op y) is defined to be the rounded version of x op y, b2 – ac > 0 implies fl(b2) – fl(ac) > 0 (rounding is a monotonic operation). The final computed answer is fl(fl(b2) - fl(ac)) = (fl(b2) - fl(ac))(1 + δ), > 0.
|δ| < u
2.18. No. A counterexample in 2-digit base-10 arithmetic is fl(2.0 – 0.91) = f l(1.09) = 1.1. 2.19. The function maps the set of positive floating point numbers onto a set of floating point numbers with about half as many elements. Hence there exist two distinct floating point numbers x having the same value of and so the condition = |x| cannot always be satisfied in floating point arithmetic. The requirement |x| is reasonable for base 2, however, and is satisfied in IEEE arithmetic, as we now show. Without loss of generality, we can assume that 1 < x < 2, since scaling x by a power of 2 does not alter By definition, is the nearest floating point number to and
Now the spacing of the floating point numbers between 1 and 2 is
= 2u, so
Hence |θ| < u if u < 1/4 (say), and then |x| is the nearest floating point number to so that In base-10 floating point arithmetic, the condition can be violated. For example, working to 5 significant decimal digits, if x = 3.1625 then fl(x2) = fl (10.0014 0625) = 10.001, and = fl(3.1624 3577 . . .) = 3.1624 < x. 2.20. On a Cray Y-MP the answer is yes, but in base-2 IEEE arithmetic the answer is no. It suffices to demonstrate that = sign(x) which is shown by the proof of Problem 2.19. 2.21. The test “x > y“ returns false if x or y is a NaN, so the code computes max(NaN, 5) = 5 and max(5, NaN) = NaN, which violates the obvious requirement that max(x, y) = max(y, x). Since the test x x identifies a NaN, the following code implements a reasonable definition of max(x, y): % max(x, y) if x x then max = y else if y y then max = x else if y > x then
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max = y else max = x end end end A further refinement is to ensure that max(-0, + 0) = +0, which is not satisfied by the code above since –0 and +0 compare as equal; this requires bit-level programming. 2.22. We give an informal proof; the details are obtained by using the model f l(x op y) = (x op y)(1 + δ), but they obscure the argument. Since a, b, and c are nonnegative, a + (b + c) is computed accurately. Since c < b < a, c + (a – b) and a + (b – c) are the sums of two positive numbers and so are computed accurately. Since a, b, and c are the lengths of sides of a triangle, a < b + c; hence, using c < b < a, b < a < b + c < 2b , which implies that a – b is computed exactly, by Theorem 2.5. Hence c – ( a – b) is the difference of two exactly represented numbers and so is computed accurately. Thus f l(A) = where is an accurate approximation to the desired argument x of the square root. It follows that fl(A) is accurate. 2.23. For a machine with a guard digit, y = x, by Theorem 2.5 (assuming 2 x does not overflow). If the machine lacks a guard digit then the subtraction produces x if the last bit of x is zero, otherwise it produces an adjacent floating point number with a zero last bit; in either case the result has a zero last bit. Gu, Demmel, and Dhillon [484, 1994] apply this bit zeroing technique to numbers d1, d2,. . . . dn arising in a divide and conquer bidiagonal SVD algorithm, their motivation being that the differences di – dj can then be computed accurately even on machines without a guard digit. 2.24. The function f(x) = 3x – 1 has the single root x = 1/3. We can have fl(f(x)) = 0 only for x 1/3. For x 1/3 the evaluation can be summarized as follows:
The first, second, and fourth subtractions are done exactly, by Theorem 2.5. The result of the first subtraction has a zero least-significant bit and the result of the
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second has two zero least-significant bits; consequently the third subtraction suffers no loss of trailing bits and is done exactly, Therefore f(x) is computed exactly for x = fl(x) near 1/3. But fl(x) can never equal 1/3 on a binary machine, so fl(f(x)) 0 for all x. 2.25. We have
where |δi | < u, i = 1:4. Hence (1 + δ4 ) = (ad – bc – bcδ 1)(1 + δ3) + bcδ1(1 + δ2) = x + xδ3 – bcδ1(δ3 – δ2), so that which implies high relative accuracy unless u|bc| >> |x|. For comparison, the bound for standard evaluation of fl(ad – bc) is |x – < γ2(|ad| + |bc|). 2.26. Newton’s method is
The quadratic convergence can be seen from 1 – xk+l a = (1 – xka ) 2 . 2.27. We would like (2.8) to be satisfied, that is, we want
= fl(x/y) to satisfy (A.1)
This implies that In place of r = – x we have to use where and where we can assume the subtraction is exact, since the case of interest is where (see Theorem 2.5). Thus using (Al) we obtain
Therefore the convergence test is
Since pute
underflows to zero it cannot be precomputed, and we should instead com-
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3.1. The proof is by induction. Inductive step: for ρ n = +l,
For ρn = – 1 we find, similarly, that
3.2. This result can be proved by a modification of the proof of Lemma 3.1. But it follows immediately from the penultimate line of the proof of Lemma 3.4. 3.3. The computed iterates
Defining
satisfy
= qk + ek, we have
This gives The running error bound µ can therefore be computed along with the continued fraction as follows: qn+1 = an+1 fn+1 = 0 for k = n: – 1:0 r k = b k /q k+l q k = a k + rk fk = |qk| + |rk| + |bk|fk+l/((|qk+l| – ufk+l|)|qk+l|) end µ = uf0 The error bound is valid provided that |qk+l| – ufk+l > 0 for all k. Otherwise a more sophisticated approach is necessary (for example, to handle the case where qk+l = 0, qk = and qk-1 is finite).
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3.4. We prove just the division result and the last result. Let Then α =
However, it is easy to verify that, in fact,
if j < k. For the last result,
3.5. We have fl(AB) = AB + ∆C = (A + ∆CB-1)B =: (A + ∆A)B, where ∆ A = ∆CB-l, and |∆C| < by (3.12). Similarly, fl(AB) = A(B + ∆B), 3.6. We have
which implies Solving these inequalities for gives the required lower bound. The definition is flawed in general as can be seen from rank considerations. For example if A and B are vectors and rank(C) 1, then no perturbations AA and ∆B exist to satisfy C = (A + ∆A)(B + ∆B)! Thus we have to use a mixed forward/backward stability definition in which we perturb C by at most as well as A and B. 3.7. Lemma 3.5 show that (3.2) holds provided that each (1 + δ )k product is replaced by xi yi . Thus we have
It is easy to show that |αι| < γ n+2, so the only change required in (3.4) is to replace γ n by γn+2. The complex analogue of (3.10) is = (A+ ∆A)x, |∆A| < γn + 2 |A|.
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3.8. Without loss of generality we can suppose that the columns of the product are computed one at a time. With xj = A1 . . . Akej we have, using (3.10), and so, by Lemma 3.6, Squaring these inequalities and summing over j yields ||A1 . . . Ak – fl(Al . . . Ak)||F < which gives the result. Note that the product ||A1||2 . . . ||Ak||2 can be much smaller than ||A||F . . . ||Ak||F; the extreme case occurs when the Ai are orthogonal. 3.9. We have fl((x + y)(x – y)) = (x + y)(x – y)(1 + θ3), |θ3| < γ3 = 3u/(1 – 3u), so the computed result has small relative error. Moreover, if y/2 < x < y then x – y is computed exactly, by Theorem 2.5, hence fl((x + y)(x – y)) = (x + y)(x – y) (1 + θ2). However, fl(x2 – y2) = x2 (1 + θ2) – y2 (1 + θ´2), so that
and we cannot guarantee a small relative error. If |x| >> |y| then fl(x2 – y2) suffers only two rounding errors, since the error in forming f1(y2) will not affect the final result, while fl((x + y)(x – y)) suffers three rounding errors; in this case fl(x2 – y2) is likely to be the more accurate result. 3.10. Assume the result is true for m – 1. Now
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3.11. The computations can be expressed as
We have
where |δi |,
< u. Solving these recurrences, we find that
It follows that which shows that however, we find that
differs negligibly from ym+l. For the repeated squarings,
where we have used Lemma 3.1. Hence the squarings introduce a relative error that can be approximately as large as 2mu. Since u = 2 –53 this relative error is of order 0.1 for m = 50, which explains the observed results for m = 50. For m = 75, the behaviour on the Sun is analogous to that on the HP calculator described in §1.12.2. On the 486DX, however, numbers less than 1 are mapped to 1. The difference is due to the fact that the 486DX uses double rounding and the Sun does not; see Problem 2.9. 3.12. The analysis is just a slight extension of that for an inner product. The analogue of (3.3) is
Hence
Setting M = max{ |f(x)| : a < x < b}, we have (A.2)
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Any reasonable quadrature rule designed for polynomic f has so one implication of (A.2) is that it is best not to have weights of large magnitude and varying sign; ideally, wi > 0 for all i (as for Gaussian integration rules, for example), so that 4.1. A condition number is C(x) = max It is easy to show that
The condition number is 1 if the xi all have the same sign. 4.2. In the (i – 1)st floating point addition the “2k-t” portion of xi does not propagate into the sum (assuming that the floating point arithmetic uses round to nearest with ties broken by rounding to an even last bit or rounding away from zero), thus there is an error of 2k-t and = i. The total error is
while the upper bound of (4.4) is
which agrees with the actual error to within a factor 3; thus the smaller upper bound of (4.3) is also correct to within this factor. The example just quoted is, of course, a very special one, and as Wilkinson [1088, 1963, p. 20] explains, “in order to approach the upper bound as closely as this, not only must each error take its maximum value, but all the terms must be almost equal.” 4.3. With S k =
we have
By repeated use of this relation it follows that
which yields the required expression for The bound on |En| is immediate. The bound is minimized if the xi are in increasing order of absolute value. This observation is common in the literature and it is sometimes used to conclude that the increasing ordering is the best one to use. This reasoning is fallacious, because” minimizing an error bound is not the same as minimizing the error itself. As (4.3) shows, if we know nothing about the signs of the rounding errors then the “best” ordering to choose is one that minimizes the partial sums.
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4.4. Any integer between 0 and 10 inclusive can reproduced. For example, fl (1 + 2 + 3 + 4 + M - M) = 0, fl(M - M + l + 2 + 3 + 4) = 10, and fl(2 + 3 + M - M + 1 + 4) = 5. 4.5. This method is sometimes promoted on the basis of the argument that it minimizes the amount of cancellation in the computation of Sn. This is incorrect: the “±” method does not reduce the amount of cancellation—it simply concentrates all the cancellation into one step. Moreover, cancellation is not a bad thing per se, as explained in §l.7. The “±’ method is an instance of Algorithm 4.1 (assuming that S+ and S– are computed using Algorithm 4.1) and it is easy to see that it maximizes max i |Ti | over all methods of this form (where, as in §4.2, Ti is the sum computed at the i t h the value of maxi |Ti | tends to be stage). Moreover, when much larger for the “±” method than for the other methods we have considered. 4.6. The main concern is to evaluate the denominator accurately when the xi are close to convergence. The bound (4.3) tells us to minimize the partial sums; these are, approximately, for xi ξ, (a) −ξ, 0, (b) 0, 0, (c) 2ξ, 0. Hence the error analysis of summation suggests that (b) is the best expression, with (a) a distant second. That (b) is the best choice is confirmed by Theorem 2.5, which shows there will be only one rounding error when the xi are close to convergence. A further reason to prefer (b) is that it is less prone to overflow than (a) and (c). 4.7. This is, of course, not a practical method, not least because it is very prone to overflow and underflow. However, its error analysis is interesting. Ignoring the error in the log evaluation, and assuming that exp is evaluated with relative error bounded by u, we have, with |δ| < u for all i, and for some δ2 n
Hence the best relative error bound we can obtain is Clearly, this method of evaluation guarantees a small absolute error, but not a small relative error when |Sn| 0 then the bounds imply
Thus (b) and (c) provide high relative accuracy for all i, while the relative accuracy of (a) can be expected to degrade as i increases. 5.1. By differentiating the Homer recurrence qi = xqi+1 + ai , qn = an, we obtain
The factors 2, 3, . . . . can be removed by redefining
Then
5.2. Analysis similar to that for Homer’s rule shows that fl(p(x)) = a0 + a1x + · · · + anxn. The total number of rounding errors is the same as for Homer’s algorithm, but they are distributed more equally among the terms of the polynomial. Homer’s rule can be expected to be more accurate when the terms |ai xi | decrease rapidly with i, such as when p(x) is the truncation of a rapidly convergent power series. Of course, this algorithm requires twice as many multiplications as Homer’s method. 5.3. Accounting for the error in forming y, we have, using the relative error counter notation (3.9),
Thus the relative backward perturbations are bounded by (3n/2 + 1)u instead of 2nu for Homer’s rule.
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5.4. Here is a MATLAB M-file to perform the task. function %LEJA % % %
[a, perm] = leja(a) LEJA ordering. [A, PERM] = LEJA(A) reorders the points A by the Leja ordering and returns the permutation vector that effects the ordering in PERM.
n = max(size(a)); perm = (l:n)’; % a(1) = max(abs(a)). [t, i] = max(abs(a)); if i ~= 1 a([1 i]) = a([i 11); perm([1 i]) = perm([i l]); end p = ones(n,l); for k = 2 : n - 1 for i = k:n p(i) = p(i)*(a(i)-a(k-1)) ; end [t, il = max(abs(p(k:n))); i = i+k-l; if i ~= k a([k i]) = a([i k]); p([k i]) = p([ik]); perm([k i]) = perm([i k]); end end 5.5. It is easy to show that the computed satisfies = p(x)(1 + θ 2 n + 1), |θ 2n+1 | < γ 2n+1. Thus has a tiny relative error. Of course, this assumes that the roots xi are known exactly! 6.1. For
then, using the Cauchy–Schwarz inequality,
The first inequality is an equality iff |aij| = α, and the second inequality is an equality iff A is a multiple of a matrix with orthonormal columns. If A is real and square, these requirements are equivalent to A being a scalar multiple of a Hadamard matrix. If A is complex and square, the requirements are satisfied by the given Vandermonde matrix, which is times a unitary matrix. 6.2.
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6.3. By the Holder inequality, (A.3) We now show that equality is possible throughout (A.3). Let x satisfy ||A|| = ||Ax||/||x|| and let y be dual to Ax. Then Re y*Ax = y*Ax = ||y||D||Ax|| = ||y||D||A|| ||x||, as required. 6.4. From (6.19) we have ||Mn||p But by taking x in the definition (6. 11) to be the vector of all ones, we see that ||Mn||P > µ n . 6.5. If A = PDQ* is an SVD then ||AB||F = ||PDQ*B||F = ||DQ*B||F = ||A||2||B||F. Similarly, ||BC||F < ||B||F||C||2, and these two inequalities together imply the required one. 6.6. By (6.6) and (6.8) it suffices to show that ||A-1||β,α = We have
6.7. Let λ be an eigenvalue of A and x the corresponding eigenvector, and form the matrix X = [x, x,. . . ,x] Then AX = showing that |λ| < ||A||. For a subordinate norm it suffices to take norms in the equation Ax = λx. 6.8. The following proof is much simpler than the usual proof based on diagonal scaling to make the off-diagonal of the Jordan matrix small (see, e.g., Horn and Johnson [580, 1985, Lem. 5.6.10]). The proof is from Ostrowski [812, 1973, Thin. 19.3]. Let δ −1 A have the Jordan canonical form δ−1 A = XJX--1. We can write where D = diag(λ i ) and the λi are the eigenvalues of A. Then A = X(D + δN)X-1, so Note that we actually have ||A|| = ρ(A) + δ if the largest eigenvalue occurs in a Jordan block of size greater than 1. If A is diagonalizable then with δ = 0 we get ||A|| = ρ(A). The last part of the result is trivial.
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6.9. Let A have the SVD A = UΣV*. By the unitary invariance of the 2- and Frobenius norms, ||A||2 = ||Σ||2 = σ1, ||A||F = ||Σ||F = Thus by ||A||2 < ||A||F < (in fact, we can replace where r = rank(A)). There is equality on the left when σ2 = · · · = σn = 0, that is, when A has rank 1 (A= xy*) or A = 0. There is equality on the right when σ1 = · · · = σ n = α, that is, when A = αQ where Q has orthonormal columns, 6.10. Let F = PΣQ* be an SVD. Then
But can be permuted into the form diag(Di ), where Di = It is easy to find the singular values of Di , and the maximum value is attained for σ1 = ||F||2. 6.11 (a)
with equality for x = ek, where the maximum is attained for j = k. (b)
Equality is attained for an x that gives equality in the Holder inequality involving the kth row of A, where the maximum is attained for i = k. Finally, from either formula, 6.12. Using the Cholesky factorization A = R*R,
6.13. HT is also a Hadamard matrix, so by the duality result (6.21) it suffices to show that ||H||P = nl/p for 1 < p < 2. Since |hij| = 1, (6.12) gives ||H||P > n1/p. Since ||H||1 = n and ||H||2 = n1/2, (6.20) gives ||H||P < nl/p, and so ||H||P = nl/p for 1 < p < 2, as required. 6.14. We prove the lower bound in (6.22) and the upper bound in (6.23); the other bounds follow on using ||AT||P = ||A||q. First, note that ||A||P > ||Aej||P = ||A(:, j)||p, which gives the lower bound in (6.22). Now assume that A has at most µ nonzeros per column. Define Di = diag(si1, . . . . sin ),
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and note that
We have
which gives the upper bound in (6.23). 6.15. The lower bound follows from ||Ax||p/||x||p < || |A||x| ||p/|| |x| ||P. From (6.12) we have
By (6.21), we also have || |A| ||P = || |AT| ||q < nl-l/q||AT||q = n1/p||A||p and the result follows. 6.16. The function v is not a vector norm because does not hold for all However, and the other two norm conditions hold, so it makes sense to define the “subordinate norm”. We have
There is equality for xj an appropriate unit vector ej. Hence v(A) = maxj v(A(:,j)). 7.1. It is straightforward to obtain from A(y – x) = ∆b – ∆Ax + ∆A(x – y) the inequality In general, if B > 0 and ρ(B) < 1 then I – B is nonsingular. Since we can premultiply by to obtain the bound for |x – y|. For the last part,
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7.2. Take norms in r = A(x–y) and x–y = A–lr. The result says that the normwise relative error is at least as large as the normwise relative residual and possibly K(A) times as large. Since the upper bound is attainable, the relative residual is not a good predictor of the relative error unless A is very well conditioned. 7.3. Let DR equilibrate the rows of A, so that B = DRA satisfies |B|e = e. Then
Hence = cond(D R A) = cond(A), which implies cond(A). The inequality cond(A) < is trivial, and the deduction of (7. 12) is immediate. 7.4. The first inequality is trivial. For the second, since hii = 1 and |hij| < 1 we have |H| > 1 and < n. Hence
7.5. We have ∆x = A-l ∆b, which yields
Now
which yields the result. If we take k = n we obtain the bound that would be obtained by applying standard perturbation theory (Theorem 7.2). The gist of this result is that the full K2(A) magnification of the perturbation will not be felt if b contains a significant component in the subspace span(U k) for a k such that This latter condition says that x must be a large-normed solution: 7.6. (a) Use upper bound, use f < |A| |x|. (b) Use
For the
7.7. We will prove the result for w; the proof for η is entirely analogous. The lower bound is trivial. Let Then and (A + ∆A)y = b + ∆b with |∆ A| < |A| and Hence |b| = |(A + ∆A)y - ∆b| < yielding Thus
which implies the result, by Theorem 7.3.
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Therefore cT∆ x = cTA-l (∆b –
this inequality being sharp. Hence
The lower bound for follows from the inequalities |cTx| = |cTA-1·Ax| < T -l T T -l |c A | |A| |x| and |c x| = |c A b| < |cTA-l ||b|. A slight modification to the derivation of (A, x) yields
7.9. (a) For any D1, D2
we have (A.4)
The rest of the proof shows that equality can be attained in this inequality. Let x1 > 0 be a right Perron vector of BC, so that BCx1 = πx1, where π = ρ(BC) > 0. Define (A.5) x2 = Cx1, so that x2 > 0 and
Bx2 = πx 1 .
(A.6)
(We note, incidentally, that x2 is a right Perron vector of CB: CBx2 = πx 2 .) Now define (A.7) D2 = diag(x 2 ). D1 = diag(x 1 ) - 1 , T or Then, with e = [1, 1, . . . , 1] , (A.6) can be written BD2e = π e. Since D1BD2 > 0, this gives similary, (A.5) can be written D2e = or = e, which gives Hence for D1 and D2 defined by (A.7) we have as required. Note that for the optimal D1 and D2, D1BD2 and both have the property that all their rows sums are equal. (b) Take B = |A| and C = |A-l|, and note that Now apply (a). (c) We can choose F1 > 0 and F2 > 0 so that |A| + tFl > 0 and |A-l| + tF2 > 0 for all i! >0. Hence, using (a),
Taking the limit as t
0 yields the result, using continuity of eigenvalues.
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(d) A nonnegative irreducible matrix has positive Perron vector, from standard Perron-Frobenius theory (see, e.g., Horn and Johnson [580, 1985, Chap. 8]). Therefore the result follows by noting that in the proof of (a) all we need is for D1 and D2 in (A.7) to be defined and nonsingular, that is, for x1 and x2 to be positive vectors. This is the case if BC and CB are irreducible (since x2 is a Perron vector of CB) (e) Using and it is easy to show that the results of (a)–(d) remain true with the co-norm replaced by the 1norm. From it then follows that inf In fact, the result in (a) holds for any p-norm, though the optimal D1 and D2 depend on the norm; see Bauer [79, 1963, Lem. 1(i)]. 7.10. That cannot exceed the claimed expression follows by taking absolute values in the expression ∆X = – (A-l ∆AA–1 + A–1 ∆A∆ X). To show it can equal it we need to show that if the maximum is attained for ( i, j) = (r,s) then can be attained, to first order in Equality is attained for ∆A = D1ED2, where D1 = diag(sign(A -l ) r i), D2 = diag(sign(A - 1 )i s ). 7.11. (a) We need to find a symmetric H satisfying Hy = b – Ay =: r. Consider the QR factorization
The constraint can be rewritten QTHQ·QTy = QTr, and it is satisfied by QTHQ := diag( H, 0 n -2) if we can find H such that where t = We can take := (||u||2/||t||2)Q, where Q is either a suitably chosen Householder matrix, or the identity if t is a positive multiple of u. Then
and (b) We can assume, without loss of generality, that |y1| < |y2| < · · · < |yn|.
(A.8)
Define the off-diagonal of H by hij = hji = gij for j > i, and let h11 = g11. The i t h equation of the constraint Hy = Gy will be satisfied if (A.9)
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If yi =0 set hii = 0; then (A.9) holds by (A.8). Otherwise, set (A.10) which yields
by the diagonal dominance. (c) Let D = and A =: Then = 1 and < 1 for i j. The given condition can be written where = Dy and = D-lb. Defining as in the proof of (b), we find from (A. 10) that so that < (2n - 1) With H := we find that H = HT, (A + H)y = b, and |H| < (2n – 1) 7.12. By examining the error analysis for an inner product (33.1), it is easy to see that for any x
Hence
satisfies (A.11)
which is used in place of (7.26) to obtain the desired bound. 7.13. (a) Writing A-1 =
we have
By the independence of the
and δι.
Hence
(b) A traditional condition number would be based on the maximum of over all “perturbations to the data of relative size σ". The expression condexp(A, b) uses the “expected perturbation” rather than the worst case. (c) For eij = ||A||2 and fi = ||b||2 we have
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so
But
Hence — This inequality shows that when perturbations are measured normwise there is little difference between the average and worst-case condition numbers. 7.14. We have
Note that this result gives a lower bound for the optimal condition number, while Bauer’s result in Problem 7.9(c) gives an upper bound. There is equality for diagonal A, trivially. For triangular A there is strict inequality in general since the lower bound is 1! 8.1. “Straightforward. 8.2. Let
Then
As This 3 x 3 example can be extended to an n x n one by padding with an identity matrix. 8.3. The bound follows from Theorem 8.9, since Using we get a similar result to that given by Theorem 8.7. 8.4. Assume T is upper triangular, and write T = D – U, where D = diag( tii ) > 0 and U > 0 is strictly upper triangular. Then, using the fact that (D-1 U)n = 0,
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Now 0 < b = TX = (D – U)x, so Dx > Ux, that is, x > D-l Ux. Hence
which gives the result, since x = T-1b > 0. 8.5. Use (8.3). 8.7. (a) Write
where Ab = y. Let |bk| =
From
it follows that
and hence that (b) Apply the result of (a) to AD and use the inequality (c) If A is triangular then we can achieve any diagonal dominance factors β ι we like, by appropriate column scaling. In particular, we can take β ι = 1 in (b), which requires M(A)d = e, where d = De. Then so the bound is as required. 8.8. (a) Using the formula det(I + xyT) = 1 + yTx we have det Hence we take if otherwise there is no αij that makes A + singular. It follows that the “best” place to perturb A to make it singular (the place that gives the smallest αij ) is in the (s, r) position, where the element of largest absolute value of A–1 is in the (r, s) position. (b) The off-diagonal elements of are given by Hence, using part (a), is singular, where α = –22–n. In fact, Tn is also made singular by subtracting 21-n from all the elements in the first column. 8.9. Here is Zha’s proof. If s = 1 the result is obvious, so assume s < 1. Define the n-vectors and
and let
It is easy to check that
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which shows that σ is a singular value of Un (θ). With σι denoting the ith largest singular value, Now we prove by induction that For n = 2 it is easy to check by direct computation that Using the interlacing property of the singular values [470, 1989, §8.3.1] and the inductive assumption, we have
Therefore 8.10. For a solver of this form, it is not difficult to see that
where denotes fi with all its coefficients replaced by their absolute values, and where (M(T), |b| ) is a rational expression consisting entirely of nonnegative terms. This is the required bound expressed in a different notation. An example (admittedly, a contrived one) of a solver that does not satisfy (8.20) is, for a 2 x 2 lower triangular system LX = b,
9.1. The proof is by induction. Assume the result is true for matrices of order n – 1, and suppose
is a unique LU factorization. Then MV = C – laT has a unique LU factorization, and so by the inductive hypothesis has nonsingular leading principal submatrices of order 1 to n – 2. Thus vii 0, i = 1:n – 2. If α = 0 then b = 0 and l is arbitrary subject to C – laT having an LU factorization. But C – ( l + ∆l)aT has nonsingular leading principal submatrices of order 1 to n – 2 for sufficiently small ∆ l (irrespective of a), so has an LU factorization for sufficiently small ∆l. Thus if α = 0 we have a contradiction to the uniqueness of the factorization. Hence α 0 , which completes the proof. 9.2. A(σ) fails to have an LU factorization without pivoting only if one of its leading principal submatrices is singular, that is, if for k {1,2, . . . ,n – l}. There are thus l + 2 + · · ·+ ( n – 1) = ½(n – 1)n “danger” values of u, which may not all be distinct. 9.3. If 0 F(A) then all the principal submatrices of A must be nonsingular, so the result follows by Theorem 9.1. Note that A may have a unique LU factorization even when 0 is in the field of values, as shown by the matrices
so the implication is only one way. Note also that 0 F(A) iff eiθ A has positive definite Hermitian part for some real θ (Horn and Johnson [581, 1991, Thin. 1.3.5]).
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the computed permutations,
and that of Theorem 9.4 becomes
9.5
9.6. Since A is nonsingular and every submatrix has a nonnegative determinant, the determinantal inequality shows that det (A (1: p, 1: p)) > 0 for p = 1: n –1, which guarantees the existence of an LU factorization (Theorem 9.1). That the elements of L and U are nonnegative follows from (9.2). GE computes since Thus For i for j Thus 0 for all i, j, k and hence = 1. 1. But 9.7. The given fact implies that JAJ is totally nonnegative. Hence it has an LU factorization JAJ = LU with L > 0 and U > 0. This means that A = (JLJ)(JUJ) is an LU factorization, and = LU = JAJ = |A|. 9.8. We start with Theorem 9.4, and so need to bound Hence
Now
implying Hence (with the same caveat as mentioned after Theorem 9.5)
9.9. By inspecting the equations in Algorithm 9.2 we see that the computed LU factors satisfy Since = b, we have, writing α = formula (Problem 24.2),
and using the Sherman–Morrison
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So
The error x – be of order
is a rational function of and is zero if xj = 0, but it will typically
9.10. α(B) = α(A), and B-l = so Taking A = Sn g(2n) >
(B) =
(A). Hence
(B) = = n +1.
9.11. First, the size or accuracy of the pivots is not the fundamental issue. The error analysis shows that instability corresponds to large elements in the intermediate matrices or in L or U. Second, in PAQ = LU, is an element of A-1 (see (9.10)), so it is not necessarily true that the pivoting strategy can force unn to be small. Third, from a practical point of view it is good to obtain small final pivots because they reveal near singularity of the matrix. 9.12. Because the rows of U are the pivot rows, µ j is the maximum number of times any element in the jth column of A was modified during the reduction. Since the multipliers are bounded by T–1, the bound for maxi follows easily. Thus p n < (1 +
T
-l maxjµj
)
.
10.1. Observe that if z = αei – ej, where ei is the ith unit vector, then for any α we have, using the symmetry of A, Since the right-hand side is a quadratic in α, the discriminant must be negative, that is, < 0, which yields the desired result. This result can also be proved using the Cholesky decomposition and the Cauchy–Schwarz inequality: A = RTR (there is strict inequality for since The inequality implies that |aij| < max(aii , ajj), which shows that that is, the largest element of A lies on the diagonal. 10.2. Compute the Cholesky factorization A = RTR, solve RTy = x, then compute yTy. In addition to being more efficient than forming A-1 explicitly, this approach guarantees a nonnegative result. 10.3. Let s = c –
Then
as required.
By Lemma 8.4,
Hence
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10.4. A= where B = C–aa T /α. Let y = [1, z] T . Then 0 < for all and so the discriminant 4(z T a)2 – T T T > 0 (since α = a11 > 0). 4αz Cz < 0 if z 0, that is, z Bz = z CZ – This shows that B is positive definite. By induction, the elimination succeeds (i.e., all pivots are nonzero) and all the reduced submatrices are positive definite. Hence > 0 for all r. Therefore
Thus a kk = > 0. Since the largest element of a positive definite matrix lies on the diagonal (Problem 10.1), for any i, j, k there exists r such that
which shows that the growth factor pn = 1 (since pn > 1). 10.5. From (10.8),
so as required. 10.6. so RZ = 0 and hence AZ = 0. Z is of dimension n (n – r) and of full rank, so it spans null(A). X
10.7. The inequalities (10. 13) follow from the easily verified fact that, for j > k,
10.8. Examples of indefinite matrices with nonnegative leading principal minors are
A necessary and sufficient condition for definiteness is that all the principal minors of A (of which there are 2n–1 ) be nonnegative (not just the leading principal minors); see, e.g., Horn and Johnson [580, 1985, p. 405] or Mirsky [763, 1961, p. 405] (same page number in both books!). 10.9. For the matrix Z = we have, from (10.15), ZTAZ = Sk (A) and so we can take p = Zel, the first column of Z.
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10.10. Theorem 10.3 is applicable only if Cholesky succeeds, which can be guaranteed only if the (suitably scaled) matrix A is not too ill conditioned (Theorem 10.7). Therefore the standard analysis is not applicable to positive semidefinite matrices that are very close to being singular. Theorem 10.14 provides a bound on the residual after rank(A) stages and, in particular, on the computed Schur complement, which would be zero in exact arithmetic. The condition of Theorem 10.7 ensures that all the computed Schur complements are positive definite, so that even if magnification of errors occurs, it is absorbed by the next part of the Cholesky factor. The proposed appeal to continuity is simply not valid. 10.11. The analysis in §10.4 shows that for a 2 x 2 pivot E, det(E) < ( α2 – l) for complete pivoting and det(E) < (α 2 – 1)λ 2 for partial pivoting. Now α2 – 1 < 0 and µ 0 and λ are nonzero if a 2 x 2 pivot is needed. Hence det(E ) |a|, so the formula det(E) = [(a/b)c – b]b minimizes the risk of overflow. Similarly, the formula
helps to avoid overflow; this is the formula used in LINPACK’S xSIDI and LAPACK’S xSYTRI. The same formulae are suitable for complete pivoting because then |b| > max( |a|, |c| ). 10.12. The partial pivoting strategy simplifies as follows: if |a11| > α|a21| use a 1 x 1 pivot all, if |a22| > α|a12| use a 1 x 1 pivot a22, else use a 2 x 2 pivot, that is, do nothing. 10.13. There may be interchanges, because the tests |a11| > αλ and αλ2 < |a11| can both fail, for example for A = with But there can be no 2 x 2 pivots, as they would be indefinite (Problem 10.11). Therefore the factorization is PAPT = LDLT for a diagonal D with positive diagonal entries. 10.14. That the growth factor bound is unchanged is straightforward to check. No 2x2 pivots are used for a positive definite matrix because, as before (Problem 10.11), any 2x2 pivot is indefinite. To show that no interchanges are required for a positive definite matrix we show that the second test, αλ2 < |a11| is always passed. The is positive definite, so a11arr – > 0. Hence submatrix
as required.
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10.15. With partial pivoting the diagonal pivoting method produces the factorization, with P = I,
As
0, ||L||
and now ||L||
In contrast, complete pivoting yields
is bounded independently of
10.16. For any nonzero x we have
Putting x1 =
we obtain
which shows that S is positive definite. 10.17. (a) The nonsingularity of A follows from the factorization
since G + BH-1 BT is symmetric positive definite. (b) It suffices to show that the first n + m – 1 leading principal submatrices of are nonsingular for any permutation matrix But these submatrices are of the form where AP is a principal submatrix of A and is a permutation matrix. Any such AP is of the form
where Hp and GP are principal submatrices of H and G, respectively, and so are positive definite. Thus AP is symmetric quasidefinite and hence nonsingular by (a), as required. (c)
so (AS + (AS)T)/2 = diag(H, G), which is symmetric positive definite.
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11.1
11.2. The inequality (11.4) yields, with x0 := 0, and dropping the subscripts on the gi and Gi ,
Now G |F| < assumed to have a modest
where B = -norm. Now, using Problem 11.1,
can be
under the conditions of Theorem 11.4. The term Gg can be bounded in a similar way. The required bound for follows. 12.1. The equations for the blocks of L and U are U11 = A11 and
First, consider the case where A is block diagonally dominant by columns. We prove by induction that which implies both the required bounds. This inequality clearly holds for k = 2; suppose it holds for k = i. We have
where
using the block diagonal dominance for the last inequality. Hence
as required. The proof for A block diagonally dominant by rows is similar. The inductive hypothesis is that
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and with P defined as before, we have
giving as required. For block diagonal dominance by columns in the -norm we have and so block LU factorization is stable. If A is block diagonally dominant by rows, stability is assured if ||Ai,i-1||/||Ai-1,i|| is suitably bounded for all a. 12.2. The block 2 x 2 matrix
is block diagonally dominant by rows and columns in the 1- and -norms for = 1/2, but is not point diagonally dominant by rows or columns. The block 2 x 2 matrix
is point diagonally dominant by rows and columns but not block diagonally dominant by rows or columns in the -norm or l-norm. 12.3. No. A counterexample is the first matrix in the solution to Problem 12.2, with = 1/2, which is clearly not positive definite because the largest element does not lie on the diagonal. 12.4. Form (12.2) it can be seen that (A-1)21 = Hence complement S =
where the Schur
S is the trailing submatrix that would be obtained after r – 1 steps of GE. It follows immediately that ||S|| < pn ||A||. For the last part, note that ||S-1|| < ||A-l||, because S–l is the (2,2) block of -1 A , as is easily seen from (12.2). 12.5. The proof is similar to that of Problem 8.7(a). We will show that 1. Let y = and let The kth equation of
gives
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Hence
which yields
in view of the column diagonal dominance, as required.
12.7. We have the block LU factorization
so that
det(X) = det(A) x det(D – CA-lB). Hence det(X) = det(AD – ACA–l B), which equals det(AD – CB) if C commutes with A. 13.2. The new bounds have norms in place of absolute values and the constants are different. 13.3. Immediate from AX – I = A(XA – I)A-1. 13.4. With 0
m and we are finished. Otherwise, for all k. By part (a) the positive sequence ||xk – a|| is monotone decreasing and therefore it converges to a limit l. Suppose that l = where δ > 0. Since β < 1, for some k we must have
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By (A.14), which is a contradiction. Hence l = α/(1 – β), as required. (c) The vectors ek can be regarded as representing the rounding errors on the kth iteration. The bound in (b) tells us that provided the error vectors are bounded by α, we can expect to achieve a relative error no larger than α/(1 – β). In other words, the result gives us an upper bound on the achievable accuracy. When this result is applied to stationary iteration, β is a norm of the iteration matrix; to make -1 β less than 1 we may have to use a scaled norm of the form ||A|| := ||XAX ||. 17.1. Let Xn be the n x n version of the upper triangular matrix
where D = diag so that ||Xn (:,j)||2 = 1, j = l:n. From (8.3), is the obvious n x n analogue of
Let A = diag(0,...,0, λ), with |λ| < 1. Then
But Therefore There is still the question of whether Xn is optimally scaled. We can perturb the zero eigenvalues of A to distinct, sufficiently small numbers so that ||Ak||2 is essentially unchanged and so that the only freedom in Xn is a diagonal scaling. Since Xn has columns of unit 2-norm, rninF=diag(fi) k2(XnF) > n–1/2 k2(Xn) (Theorem 7.5), so even for the optimally scaled Xn the bound can be arbitrarily poor. 17.2. A simple example is
for which λ(A) = {–1,1,3} and λ(|A|) = {–l,2±(4α+l)1/2}, so p(|A|)/p(A)
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18.1. For the Householder matrix I – (2/vTv)vvT there is an eigenvalue –1 with eigenvector v, and there are n – 1 eigenvalues 1, with eigenvectors any basis for span(v). A Givens matrix G(i,j,θ) has eigenvalues e± iθ, together with n – 2 eigenvalues 1. 18.2. Straightforward manipulation shows that a bound holds of the form cnu + 0(u2), where c is a constant of order 10. 18.3. We must have x*Px = x*y, so x*y must be real. Also, we need x*x = y*y and x y. Then, with v = x – y, v*v = 2v*x, so Px = y. LAPACK uses modified Householder matrices P = I – tvv* that are unitary but not Hermitian (T is not real). The benefit of this approach is that such a P can always be chosen so that Px = αe1, with α real (for this equation to hold for a genuine Householder matrix P it would be necessary that x1 be real). For more details see Lehoucq [698, 1994]. 18.4. False. det(P) = – 1 for any Householder matrix and det(G) = 1 for any Givens matrix, so det(Q) = 1. Moreover, whereas P is generally a full, symmetric matrix, Q has some zero entries and is generally nonsymmetric. 18.5. The inequalities follow from the fact that, in the notation of (18.1), ||xk||2 = |rkk| and ||ck(:,j)|| = the latter equality being a consequence of the invariance of the 2-norm under orthogonal transformations. The last part follows because QR factorization with column pivoting on A is essentially equivalent to Cholesky factorization with complete pivoting applied to ATA. 18.6. If y = |W|x and W comprises r disjoint rotations then, in a rather loose not at ion,
18.7. Straightforward. This problem shows that the CGS and MGS methods correspond to two different ways of representing the orthogonal projection onto span{q1, . . . ,qj}. 18.8. Assume, without loss of generality, that ||a1||2 < ||a2||2. If E is any matrix such that A + E is rank deficient then We take E = [el, 0], where el is chosen so that = 0 and al + el = αa2, for some α. From Pythagoras’s theorem we have that ||el||2 = tan θ ||al||2, and so Together with the trivial bound the result.
this yields
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18.9. We find that
for i j, but for It is easy to see = 1 implies Hence for and that showing, as expected, that the loss of orthogonality for MGS is bounded in terms of k2(A)u
For that
18.10. P is the product Pn . . . P1, where Pk is defined in (18.20). Since = 0, we have
Hence
This is of the required form, with Q = [q1,.. ., qn] the matrix from the MGS method applied to A. 18.11. With Q defined as in the hint, ∆ A= = = = =
QR– A=(Q–P21)R+∆A 2 T (VW – P21)R + ∆A2 V(I – S)WTR + ∆Az V(I + S)-1 C2 WTR + ∆A2 V(I + S)-l WT · WCUT · UCWT · R + ΑAz
and the result follows. 18.12. To produce the Householder method’s P we have to explicitly form the product of the individual Householder transformations. As long as this is done in a stable way the computed P is guaranteed to be nearly orthogonal. MGS’s Q is formed in an algebraically different way, and the rounding errors in its formation are different from those in the formation of P; in other words, Q is not a submatrix of P. Consequently, there is no reason to expect Q to be nearly orthonormal. Further insight can be gained by a detailed look at the structure of the computed P; see Björck and Paige [119, 1992, §4].
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18.13. It is straightforward to show that ATA– I = (A – U)T(A + U). Taking norms gives the lower bound. Since A + U = U(H + I) we have, from the previous relation, (A - U)TU = (ATA - I)(H + I)-1. Hence
In fact, the result holds for any unitarily invariant norm (but the ||A||2 + 1 in the denominator must be retained). 19.1. One approach is to let x be a solution and y an arbitrary vector, and consider f (α) = ||A(x + αy) – b|| By expanding this expression it can be seen that if the normal equations are not satisfied then α and y can be chosen so that f(α) < f(0). Alternatively, note that for f(z) = (b – Ax)T(b – Ax) = xTATAx – 2bTAx + bTb we have f(x) = 2(ATAx – ATb) and 2f(x) = 2ATA, so any solution of the normal equations is a local minimum of f, and hence a global minimum since f is quadratic. The normal equations can be written as (b – Ax)TA = 0, which shows that the residual b – Ax is orthogonal to range(A). 19.2. A = Q [~]. The computed upper triangular QR factor satisfies A + ∆A =
with |∆A| < mnγ cmG1|Al and ||G1||F = 1, by Theorem 18.4. BY
Lemma 18.3 the computed transformed right-hand side satisfies = QT(b + ∆b), with |∆b| < mnγ cmG1 |b|. By Theorem 8.5, the computed solution to the triangular system satisfies So
exactly solves the LS problem
which is equivalent to the LS problem, on premultiplying by Q,
We have
where G > G1 and ||G||F = 1. The normwise bounds are proved similarly. 19.3. A straightforward verification.
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19.4. Notice that
which is minimized if ||Axi — ei ||2 is minimized for i = 1: m. Thus we have m + independent LS problems. The solution is xi = A ei , i = 1: m, that is, X = + + A Im = A . This is the unique solution only if A has full rank, but it is always the unique minimum Frobenius norm solution. 19.5. By Theorem 18.12 there is an orthonormal matrix [W1, wn+l] such that
with
The computed solution satisfies
Therefore
exactly solves the LS problem
which, on premultiplying by [W1, wn+1], is equivalent to the LS problem
We have
where
and
= 1. Normwise bounds on
19.6. Subtracting
from
and
follow similarly.
we have
by (19.12). Taking norms gives the result.
to
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19.7. By constructing a block LU factorization it is straightforward to show that det(C(α) – λI) = (α – λ)m-n det(ATA + λ(α – λ)). Hence the eigenvalues of C(α) are λ = α (m – n times) together with the solutions of λ(α – λ) = namely, Now
so the minimum condition number must occur when gives for which
This
The lower bound for the maximum is achieved for 19.8. Let y = 0 and ||(A + ∆A)y – b||2 = min. If b = 0 then we can take AA= O. Otherwise, the normal equations tell us that (A+ ∆A)Tb = 0, so ||∆ A||2 > ||ATb||2/||b||2. This lower bound is achieved, and the normal equations satisfied, for ∆AT = –(ATb)bT/bTb. Hence
19.9. For the case λ∗ < 0 we have
20.1. Setting the gradient of (20.13) to zero gives ATAx – ATb + c = 0, which can be written as y = b – Ax, ATy = c, which is (20.12). The Lagrangian for (20.14) is L(y, x) = ½(y – b)T(y – b) + xT(ATy – c). yL(y, x) = y – b + Ax, and T xL(y, x) = A y – c. Setting the gradients to zero gives the system (20.12). 21.1. The modified version has the same flop count as the original version. 21.4. The summation gives
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which implies the desired equality. It follows that all columns of V–1 except the first must have both positive and negative entries. In particular, V-1 > 0 is not possible. The elements of V-1 sum to 1, independently of the points αi (see also Problem 13.7). 21.5. We have U(i,:)T(αo,. . . ,αn) = W(i,:)V(αo,. . . ,αn)- But T = LV, where L has the form illustrated for n = 5 by
T -T T -T and Le = e, so U(i,:)L = W(i,:), or U(i,:) = L W(i,:) . But L > 0 -T -1 -l n by the given x formula, so ||L ||1 = ||L || = ||L e|| = ||e|| = 1, hence ||U(i,:)||1 < ||W(i,:)||1. As an aside, we can evaluate ||L|| as
after a little trigonometric algebra. 21.7. Denote the matrix by C. For the zeros of Tn+1 we have
It follows that Extrema of Tn: we have
giving the result.
CDCT = Hence B = CD½ satisfies BBT = k2 (C) < k2(D½)k2 (B) = 2.
and so k 2 (B) = K 2 (D) ½ =
Then
21.8. The increasing ordering is never produced, since the algorithm must choose α1 to maximize |α l – α0|. 21.10. The dual condition number is
See Higham [533, 1987] for proofs.
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22.1. Consider the case where m < min(n,p), and suppose n = jm and p = km for some integers j and k. Then the multiplication of the two matrices can be split into m x m blocks:
which involves a total of jk multiplications of m x m matrices, each involving O(mα ) operations. Thus the total number of operations is O(jkmα ), or O(mα -2np), as required, and we can show similar results for the cases when n and p are the smallest dimensions. 22.2. ½n3 + n2 multiplications and ³/2n3 + 2n(n – 1) additions. 22.3. For large n = 2k, Sn (8)/Sn (n) 1.96 x (7/8) k and S n (l)/S n (8) 1.79. The ratio Sn(8)/Sn(n) measures the best possible reduction in the amount of arithmetic by using Strassen’s method in place of conventional multiplication. The ratio S n (1)/S n (8) measures how much more arithmetic is used by recurring down to the scalar level instead of stopping once the optimal dimension n0 is reached. Of course, the efficiency of a practical code for Strassen’s method also depends on the various non-floating-point operations. 22.5. Apart from the differences in stability, the key difference is that Winograd’s formula relies on commutativity and so cannot be generalized to matrices. 22.7. Some brief comments are given by Douglas, Heroux, Slishman, and Smith [317, 1994]. 22.9. The inverse is
Hence we can form AB by inverting a matrix of thrice the dimension. This result is not of practical value, but it is useful for computational complexity analysis. 24.2. With n = 3 and almost any starting data, the backward error can easily be made of order 1, showing that the method is unstable. However, the backward error is found to be roughly of order so the method may have a backward error bound proportional to (this is an open question). 25.1. (b) An optimizing compiler might convert the test xp1 > 1 to x+1 > 1 and then to x > 0. (For a way to stop this conversion in Fortran, see the solution to Problem 25.3.) Then the code would compute a number of order 2 e min instead of a number of order 2–t.
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25.3. The algorithm is based on the fact that the positive integers that can be exactly represented are 1,2, ..., β t and
In the interval [β t, β t+l] the floating point numbers are spaced β apart. This interval k must contain a power of 2, a = 2 . The first while loop finds such an a (or, rather, the floating point representation of such an a) by testing successive powers 2i to see if both 2i and 2i + 1 are representable. The next while loop adds successive powers of 2 until the next floating point number is found; on subtracting a the base β is produced. Finally t is determined as the smallest power of β for which the distance to the next floating point number exceeds 1. The routine can fail for at least two reasons. First, an optimizing compiler might simplify the test while (a+l) - a == 1 to while 1 == 1, thus altering the meaning of the program. Second, in the same test the result (a+l) might be held in an extra length register and the subtraction done in extra precision. The computation would then reflect this higher precision and not the intended one. We could try to overcome this problem by saving the result a+l in a variable, but an optimizing compiler might undo this effort by dispensing with the variable and storing the result in a register. In Fortran, the compiler’s unwanted efforts can be nullified by a test of the form while foo (a+l) - a == 1, where foo is a function that simply returns its argument. The problems caused by the use of extra length registers were discussed by Gentleman and Marovich [437, 1974]; see also Miller [759, 1984, §2.2]. 25.4. (This description is based on Schreiber [903, 1989].) The random number generator in matgen repeats after 16384 numbers [668, 1981, p. 19]. The dimension n = 512 divides the period of the generator (16384 = 512 x 32), with the effect that the first 32 columns of the matrix are repeated 16 times (512 = 32 x 16), so the matrix has this structure: B = rand(512,32); A = [B, B, B, B, B, B, B, B, B, B, B, B, B, B, B, B]; Now consider the first 32 steps of Gaussian elimination. We apply 32 transformations to A that have the effect, in exact arithmetic, of making B upper trapezoidal. In floating point arithmetic, they leave a residue of small numbers (about u 10-7 in size) in rows 33 onward. Because of the structure of the matrix, identical small numbers occur in each of the 15 blocks of A to the right of the first. Thus, the remaining (512 – 32) x (512 – 32) submatrix has the same block structure as A (but with 15 block columns). Hence this process repeats every 32 steps: after 32 steps the elements drop to 10 -7 ; after 64 steps the elements drop to 10 –14 ; after 96 steps the elements drop to 10 -21 ; after 128 steps the elements drop to 10 –28 ; after 160 steps the elements drop to 10 –35 ; after 192 steps the elements would drop to 10 –42 ,
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but that is less than the under flow threshold. The actual pivots do not exactly match the analysis, which is probably due to rank deficiency of one of the submatrices generated. Also, underflow occurs earlier than predicted, apparently because two small numbers (both O( 10–21 ) ) are multiplied in a saxpy operation. 25.5. Let si and tt denote the values of s and t at the start of the ith stage of the algorithm. Then (A.15) If |xi | > ti then the algorithm uses the relation
so with si+l = (ti /xi )2si + 1 and ti+l = |xi|, (A.15) continues to hold for the next value of i. The same is true trivially in the case |xi| < ti . This is a one-pass algorithm using n divisions that avoids overflow except, poswhich can overflow only if ||x||2 does. sibly, on the final stage in forming 25.7. (a) The matrix A := I + Y2 = I – YTY is symmetric positive semidefinite since ||Y||2 < 1, hence it has a (unique) symmetric positive semidefinite square root X satisfying X2 = A = I + Y2. The square root X is a polynomial in A (see, for example, Higham [532, 1987]) and hence a polynomial in Y; therefore X and Y commute, which implies that (X+ Y)T(X + Y) = (X – Y)(X + Y) = X2 – Y2 = I, as required. 25.8. |x|/3 could underflow to zero, or become unnormalized, when not.
does
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Appendix B
Singular Value Decomposition, M-Matrices
The singular value decomposition is a matrix factorization which can produce approximations to large arrays. Cryptanalysts is the task of breaking coded messages. In this paper, we present an unusual merger of the two in which the singular value decomposition may aid the cryptanalyst in discovering vowels and consonants in messages coded in certain variations of simple substitution ciphers. — CLEVE B. MOLER and DONALD MORRISON, Singular Value Analysis of Cryptograms (1983)
then each of the following 50 conditions is equivalent to the statement: “A is a nonsingular M-matrix”. — ABRAHAM BERMAN and ROBERT J. PLEMMONS, Nonnegative Matrices in the Mathematical Sciences (1994)
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In this appendix we define the singular value decomposition and M-matrices.
B.1. Singular Value Decomposition Any matrix
has a singular value decomposition (SVD)
are both unitary. The are the singular values of A and the columns of U and V are the left and right singular vectors of A, respectively. For more details on the SVD see Golub and Van Loan [470, 1989, pp. 7173], Stewart and Sun [954, 1990, pp. 30-34], and Horn and Johnson [580, 1985, §7.3], [581, 1991, §3.1]. The history of the SVD is described by Stewart [950, 1993] and Horn and Johnson [581, 1991, $3.0].
B . 2 . M-Matrices and all the A matrix is an M-matrix if aij < 0 for all eigenvalues of A have nonnegative real part. This is one of many equivalent definitions [94, 1994, Chap. 6]. An M-matrix may be singular. A particularly useful characterization of a nonsingular M-matrix is a nonsingular matrix for which aij < 0 for all and A-1 has nonnegative elements (written as A–1 > 0). For more information on ill-matrices see Berman and Plemmons [94, 1994] and Horn and Johnson [581, 1991, §2.5].
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Appendix C
Acquiring Software
Caveat receptor . . . Anything free comes with no guarantee! — JACK DONGARRA and ERIC GROSSE, Netlib mail header
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In this appendix we provide information on how to acquire software mentioned in the book. First, we describe some basic aspects of the Internet.
C. 1. Internet A huge variety of information and software is available over the Internet, the worldwide combination of interconnected computer networks. The location of a particular object is specified by a URL, which stands for “(Uniform Resource Locator”. Examples of URLS are http://www.netlib.erg/index.html ftp://ftp.netlib. org The first example specifies a World Wide Web server (http = hypertext transfer protocol) together with a file in hypertext format (html = hypertext markup language), while the second specifies an anonymous ftp site. In any URL, the site address may, optionally, be followed by a filename that specifies a particular file. For more details about the Internet see on-line information, or one of the many books on the subject, such as Krol [674, 1994].
C.2. Netlib Netlib is a repository of freely available mathematical software, documents, and databases of interest to the scientific computing community [316, 1987], [151, 1994]. It includes l
research codes,
l
golden oldies (classic programs that are not available in standard libraries),
l
the collected algorithms of the ACM,
l
program libraries such as EISPACK, LINPACK, LAPACK, and MINPACK
l
back issues of NA-Digest, a weekly digest for the numerical analysis community,
l
databases of conferences and performance data for a wide variety of machines.
Netlib also enables the user to download technical reports from certain institutions, to download software and errata for textbooks, and to search “the SIAM membership list and a “white pages” database. Netlib can be accessed in several ways.
C.3 M ATLAB
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l
Over the World Wide Web. The URL is http://www.netlib. erg/index.html
l
Via Xnetlib, an X Windows application that provides interactive’ access to netlib.
l
By anonymous ftp. The URL is ftp://ftp.netlib.org
l
By electronic mail. For an introduction and master index, send a oneline email message as follows: mail netlib@ornl. gov send index
Netlib is mirrored at various sites throughout the world.
C.3. M ATLAB M ATLAB is a commercial program sold by The MathWorks, Inc. It runs on a variety of platforms. The MathWorks maintains a collection of usercontributed M-files, which is accessible over the Internet. For information contact The MathWorks, Inc. 24 Prime Park Way Natick, MA 01760-1500 USA Tel: 5086531415 Fax: 5086532997 email:
[email protected] URL: ftp://ftp.mathworks.com URL: http://www.mathworks.com
C.4. NAG Library and FTN90 Compiler The Numerical Algorithms Group (NAG) produces a variety of software products. Relevant to this book are the FTN90 compiler and the NAG Library, a large numerical program library available in several programming languages. For information contact NAG Ltd. Wilkinson House
A CQUIRING S OFTWARE
584 Jordan Hill Road Oxford, OX2 8DR UK Tel: +44 1865 511245 Fax: +44 1865 310139 email:
[email protected] URL: http://www.nag.co.uk:70/
NAG has subsidiaries and distributors, whose addresses can be obtained from the above sources.
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Appendix D
Program Libraries
Since the programming is likely to be the main bottleneck in the use of an electronic computer we have given a good deal of thought to the preparation of standard routines of considerable generality for the more important processes involved in computation. By this means we hope to reduce the time taken to code up large-scale computing problems, by building them up, as it were, from prefabricated units. — J. H. WILKINSON, The Automatic Computing Engine at the National Physical Laboratory (1948)
In spite of the self-contained nature of the linear algebra field, experience has shown that even here the preparation of a fully tested set of algorithms is a far greater task than had been anticipated. — J. H. WILKINSON and C. REINSCH, Handbook for Automatic Computation: Linear Algebra (1971)
585
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In this appendix we briefly describe some of the freely available program libraries that have been mentioned in this book. These packages are all available from netlib (see §C.2).
D.1. Basic Linear Algebra Subprograms The Basic Linear Algebra Subprograms (BLAS) are sets of Fortran primitives for matrix and vector operations. They cover all the common operations in linear algebra. There are three levels, corresponding to the types of object operated upon. In the examples below, x and y are vectors, A, B, C are rectangular matrices, and T is a square triangular matrix. Names of BLAS routines are given in parentheses. The leading “x” denotes the Fortran data type, whose possible values are some or all of S D C Z
real double precision complex complex* 16, or double complex
Level 1: [694, 1979] Vector operations. Inner product: x T y (xdot); y αx + y (xAXPY); vector 2-norm (yTy)1/2 (xNRM2); swap vectors x y (xSWAP); scale a vector x αx (xSCAL); and other operations. Level 2: [313, 1988], [314, 1988] Matrix-vector operations. Matrix times vector (gaxpy): y αAx + βy (xGEMV); rank-1 update: A A + αx y T (xGER); triangular solve: x T -lx (xTRSV); and variations on these. Level 3: [308, 1990], [309, 1990] Matrix-matrix operations. Matrix multiplication: C αAB + βC (xGEMM); multiple right-hand side triangular solve: A αT -1A (xTRSM); rank-r and rank-2r updates of a symmetric matrix (xSYRK, xSYR2K); and variations on these. The BLAS are intended to be called in innermost loops of linear algebra codes. Usually, most of the computation in a code that uses BLAS calls is done inside these calls. LINPACK [307, 1979] uses the level-l BLAS throughout (model Fortran implementations of the level-l BLAS are listed in [307, 1979, App. D]). LAPACK [17, 1995] exploits all three levels, using the highest possible level at all times. Each set of BLAS comprises a set of subprogram specifications only. The specifications define the parameters to each routine and state what the routine must do, but not how it must do it. Thus the implementor has freedom over the precise implementation details (loop orderings, block algorithms, special code for special cases) and even the method (fast versus conventional matrix multiply), but the implementation is required to be numerically stable, and
D.2 EISPACK
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code that tests the numerical stability is provided with the model implementations [309, 1990], [314, 1988]. For more details on the BLAS and the advantages of using them, see the defining papers listed above, or, for example, [315, 1991] or [470, 1989, Chap. 1].
D.2. EISPACK EISPACK is a collection of Fortran 66 subroutines for computing eigenvalues and eigenvectors of matrices [925, 1976], [415, 1977]. It contains 58 subroutines and 13 drivers. The subroutines are the basic building blocks for eigensystem computations; they perform such tasks as reduction to Hessenberg form, computation of some or all of the eigenvalues/vectors, and back transformations, for various types of matrix (real, complex, symmetric, banded, etc.). The driver subroutines provide easy access to many of EISPACK’s capabilities; they call from one to five other EISPACK subroutines to do the computations. EISPACK is primarily based on Algol 60 procedures developed in the 1960s by 19 different authors and published in the journal Numerische Mathematik. An edited collection of these papers was published in the Handbook for Automatic Computation [1102, 1971].
D.3. LINPACK LINPACK is a collection of Fortran 66 subroutines that analyse and solve linear equations and linear least squares problems [307, 1979]. The package solves linear systems whose matrices are general, banded, symmetric indefinite, symmetric positive definite, triangular, or tridiagonal. In addition, the package computes the QR and singular value decompositions and applies them to least squares problems. All the LINPACK routines use calls to the level-l BLAS in the innermost loops; thus most of the floating point arithmetic in LINPACK is done within the level-l BLAS.
D.4. LAPACK LAPACK [17, 1995] was released on February 29, 1992. As the announcement stated, “LAPACK is a transportable library of Fortran 77 subroutines for solving the most common problems in numerical linear algebra: systems of linear equations, linear least squares problems, eigenvalue problems, and singular value problems. It has been designed to be efficient on a wide range of modern high-performance computers.”
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LAPACK has been developed over a period that began in 1987 by a team of 11 numerical analysts in the UK and the USA. LAPACK can be regarded as a successor to LINPACK and EISPACK; it has virtually all their capabilities and much more besides. LAPACK improves on LINPACK and EISPACK in four main respects: speed, accuracy, robustness, and functionality. It was designed at the outset to exploit the level-3 BLAS. Development of LAPACK continues under the auspices of two follow-on projects, LAPACK 2 and ScaLAPACK. An object-oriented C++ extension to LAPACK has been produced, called LAPACK++ [311, 1995]. CLAPACK is a C version of LAPACK, converted from the original Fortran version using the f2c converter [367, 1990]. ScaLAPACK comprises a subset of LAPACK routines redesigned for distributed memory parallel machines [206, 1992], [205, 1994]. Other work includes developing codes that take advantage of the careful rounding and exception handling of IEEE arithmetic [298, 1994]. For more details of all these topics see [17, 1995]. LAPACK undergoes regular updates, which are announced on the electronic newsletter NA-Digest. At the time of writing, the current release is version 2.0, dated September 30, 1994, and the package contains over 1000 routines and over 735,000 lines of Fortran 77 code, including testing and timing code. Mark 16 onward of the NAG Fortran 77 Library contains much of LAPACK in Chapters F07 and F08.
D.4.1. Structure of LAPACK The LAPACK routines can be divided into three classes. The drivers solve a complete problem. The simple drivers have a minimal specification, while the expert drivers have additional capabilities of interest to the sophisticated user. The computational routines perform individual tasks such as computing a factorization or reducing a matrix to condensed form; they are called by the drivers. The auxiliary routines perform relatively low-level operations such as unblocked factorization, estimating or computing matrix norms, and solving a triangular system with scaling to prevent overflow. The driver and computational routines have names of the form xyyzzz. The first letter specifies the data type, which is one of S, D, C, and Z. The second two letters refer to the type of matrix. A partial list of types is as follows (there are 27 types in all):
D.4 LAPACK BD GB GE GT HS OR PO PT SB ST SY TR
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bidiagonal general band general general tridiagonal upper Hessenberg (real) orthogonal symmetric or Hermitian positive definite symmetric or Hermitian positive definite tridiagonal (real) symmetric band (real) symmetric tridiagonal symmetric triangular (or (quasi-triangular)
The last three characters specify the computation performed. TRF TRS CON
RFS
TRI EQU
factorize solve a (multiple right-hand side) linear system using the factorization estimate 1/k1(A) (or compute it exactly when A is tridiagonal and symmetric positive definite or Hermitian positive definite) apply fixed precision iterative refinement and compute the componentwise relative backward error and a forward error bound use the factorization to compute A–1 compute factors to equilibrate the matrix
The auxiliary routines follow a similar naming convention, with most of them having yy = LA.
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Appendix E
The Test Matrix Toolbox The Test Matrix Toolbox is a collection of MATLAB M-files containing test matrices, routines for visualizing matrices, routines for direct search optimization, and miscellaneous routines that provide useful additions to MATLAB’S existing set of functions. There are over 50 parametrized test matrices, which are mostly square, dense, nonrandom, and of arbitrary dimension. The test matrices include ones with known inverses or known eigenvalues; ill-conditioned or rank deficient matrices; and symmetric, positive definite, orthogonal, defective, involuntary, and totally positive matrices. The visualization routines display surface plots of a matrix and its (pseudo-) inverse, the field of values, Gershgorin disks, and two- and three-dimensional views of pseudospectra. The direct search optimization routines implement the alternating directions method, the multidirectional search method, and the Nelder–Mead simplex method (which are described in $24.2). Among the miscellaneous routines are one for rounding matrix elements to a specified number of bits and ones that implement the classical and modified Gram-Schmidt methods and LU factorization without pivoting and with complete pivoting. The Test Matrix Toolbox was originally released in 1989, and a version was published as ACM Algorithm 694 [548, 1991]. The current version, version 3.0, is described in the manual [561, 1995], which should be consulted for further details. The Test Matrix Toolbox is available by anonymous ftp from The MathWorks; the URL is ftp://ftp.mathworks.com/pub/contrib/linalg/testmatrix The manual [561, 1995] is testmatrix.ps in the same location. We summarize the contents of the toolbox in the following tables, which list the M-files by category, with short descriptions.
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Demonstration tmtdemo Demonstration of Test Matrix Toolbox.
Test Matrices, A–K augment cauchy chebspec chebvand chow circul clement compan condex cycol dingdong dorr dramadah fiedler forsythe frank gallery gearm gfpp grcar hadamard hanowa hilb invhess invol ipjfact jordbloc kahan kms krylov
Augmented system matrix. Cauchy matrix. Chebyshev spectral differentiation matrix. Vandermonde-like matrix for the Chebyshev polynomials. Chow matrix—a singular Toeplitz lower Hessenberg matrix. Circulant matrix. Clement matrix—tridiagonal with zero diagonal entries. Companion matrix. “Counterexamples” to matrix condition number estimators. Matrix whose columns repeat cyclically. Dingdong matrix—a symmetric Hankel matrix. Dorr matrix—diagonally dominant, ill conditioned, tridiagonal. A (O, 1) matrix whose inverse has large integer entries. Fiedler matrix—symmetric. Forsythe matrix—a perturbed Jordan block. Frank matrix—ill conditioned eigenvalues. Famous, and not so famous, test matrices. Gear matrix. Matrix giving maximal growth factor for Gaussian elimination with partial pivoting. Grcar matrix—a Toeplitz matrix with sensitive eigenvalues. Hadamard matrix. A matrix whose eigenvalues lie on a vertical line in the complex plane. Hilbert matrix. Inverse of an upper Hessenberg matrix. An involutory matrix. A Hankel matrix with factorial elements. Jordan block. Kahan matrix—upper trapezoidal. Kac–Murdock–Szego Toeplitz matrix. Krylov matrix.
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wilk
Test Matrices. L–Z Lauchli matrix—rectangular. Lehmer matrix-symmetric positive definite. A tridiagonal matrix with real, sensitive eigenvalues. Lotkin matrix. A matrix with given Jordan canonical form. Symmetric positive definite matrix min( i , j ) . Moler matrix-symmetric positive definite. Singular matrix from the discrete Neumann problem (sparse). Random, orthogonal upper Hessenberg matrix. Orthogonal and nearly orthogonal matrices. Parter matrix—a Toeplitz matrix with singular values near p. Pascal matrix. Symmetric positive definite Toeplitz matrix. Pei matrix. Pentadiagonal Toeplitz matrix (sparse). Block tridiagonal matrix from Poisson’s equation (sparse). Prolate matrix—symmetric, ill-conditioned Toeplitz matrix. Random matrix with elements – 1, 0, or 1. Random matrix with pre-assigned singular values. A (0,1) matrix of Redheffer associated with the Riemann hypothesis. A matrix associated with the Riemann hypothesis. An upper quasi-triangular matrix. Smoke matrix-complex, with a “smoke ring” pseudospectrum. Tridiagonal matrix (sparse). Upper triangular matrix discussed by Wilkinson and others. Vandermonde matrix. Wathen matrix—a finite element matrix (sparse, random entries). Various specific matrices devised/discussed by Wilkinson.
fv gersh ps pscont see
Field of values (or numerical range). Gershgorin disks. Dot plot of a pseudospectrum. Contours and colour pictures of pseudospectra. Pictures of a matrix and its (pseudo-) inverse.
lauchli lehmer lesp lotkin makejcf minij moler neumann ohess orthog parter pascal pdtoep pei pentoep poisson prolate rando randsvd redheff riemann rschur smoke tridiag triw vend wathen
Visualization
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Decompositions and Factorization Classical Gram–Schmidt QR factorization. Cholesky factorization with pivoting of a positive semidefinite matrix. cod Complete orthogonal decomposition. diagpiv Diagonal pivoting factorization with partial pivoting. ge Gaussian elimination without pivoting. gecp Gaussian elimination with complete pivoting. Gauss-Jordan elimination to solve Ax = b. gj mgs Modified Gram–Schmidt QR factorization. poldec Polar decomposition. signm Matrix sign decomposition. cgs cholp
Direct Search Optimization adsmax mdsmax mmsmax
Alternating directions direct-search method. Multidirectional search method for direct search optimization. Nelder–Mead simplex method for direct search optimization.
Miscellaneous Band reduction by two-sided unitary transformations. Round matrix elements. Comparison matrices. Matrix condition number in 1, 2, Frobenius, or -norm. Determine suitable axis for plot of complex vector. Dual vector with respect to Holder p-norm. Eigenvalue condition numbers. Householder matrix. Test Matrix Toolbox information and matrix access by number. matsignt Matrix sign function of a triangular matrix. pnorm Estimate of matrix p-norm (1 < p < ). qmult Pre-multiply by random orthogonal matrix. rq Rayleigh quotient. seqa Additive sequence. seqcheb Sequence of points related to Chebyshev polynomials. seqm Multiplicative sequence. show Display signs of matrix elements. skewpart Skew-symmetric (skew-Hermitian) part. sparsify Randomly sets matrix elements to zero. sub Principal submatrix. symmpart Symmetric (Hermitian) part. trap2tri Unitary reduction of trapezoidal matrix to triangular form. bandred chop comp cond cpltaxes dual eigsens house matrix
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Name Index Science is organized knowledge. — HERBERT SPENCER, Essays On Education (1861)
A suffix “t” after a page number denotes a table, “f” a figure, “n” a footnote, and “q” a quotation.
Axelsson, Owe, 343
Aasen, Jan Ole, 226 Abdelmalek, Nabih N., 385, 410 Abramowitz, Milton, 35 Acton, Forman S., 35, 203 q, 285 q Adams, Duane A., 113 Aggarwal, Vijay B., 84 Ahac, Alan A., 198 Ahlberg, J. H., 167 Aitken, A. C., 383 Albers, Donald J., 513 n Alefeld, Göltz, 486 Alexopolous, Aristides G., 54 Allen, Jr., Richard C., 84 Almacany, Montaha, 425 q Aluru, Srinivas, 459 Alvarado, Fernando L., 165 Amato, James J., 323 Ames, William F., 32 Amodio, Pierluigi, 197 Anda, Andrew A., 385 Anderson, E., 289 q Anderson, T. W., 527 Ando, T., 196 Arioli, Mario, 143, 304, 328, 343, 410, 411, 421, 422 Arnold, William F., 323 Ashenhurst, R. L., 489 Asplund, Edgar, 305 Atanasoff, John V., 151 q
Ivo, 100 Bachelis, Boris, 61 Bai, Zhaojun, 322, 352, 527 Bailey, David H., 448, 460, 461, 471, 482, 491 q, 505 Baksalary, J. K., 323, 324 Bane, Susanne M., 461 Ballester, C., 441 Bank, Randolph E., 257 Bareiss, E. H., 53, 191 Bargmann, V., 186 Barlow, Jesse L., 33, 51, 53, 191, 193, 301, 385, 411, 412 Barnett, S., 323 Barone, John L., 391 q, 410 Barrett, Geoff, 59 Barrlund, Anders, 194, 198, 224 Bartels, R. H., 312 Bartels, Sven G., 141, 304, 441 Barwell, Victor, 226 Bauer, F. L., 57, 83, 119, 126, 139, 146, 148, 149, 191, 284, 551 Beam, Richard M., 527 Beaton, Albert E., 391 q, 410 Bell, E. T., 281 q Bellman, Richard, 324 Benford, Frank, 51 Benôit, Commandant, 224 665
Next
666 Benschop, N. F., 327 Berman, Abraham, 147, 579 q, 580 Bhatia, Rajendra, 322, 382 Bini, Dario, 441, 456, 457 Birkhoff, Garrett, 32, 281, 483 Bischof, Christian H., 299, 301, 370, 384 Björck, Åke, 83, 242, 361q, 379, 385, 386, 388, 397, 399, 400, 402, 403, 409-411, 422, 423, 434, 436, 441, 570 Bjørstad, Petter, 461 Blanch, G., 507 Bliss, B., 488 Blue, James L., 502 Bodewig, E., 188 Bohlender, Gerd, 98 Bohte, Z., 183 Boley, Daniel, 243 Bollen, Jo A. M., 328 Bondeli, S., 525 Boros, T., 441 Borwein, J. M., 491 q Borwein, P. B., 491 q Bowden, B. V., 203q Bowdler, H. J., 196 Boyd, David W., 291, 293, 304 Boyle, Jeff, 51 Brent, Richard P., 51, 162, 164, 448, 451, 453, 460, 486, 504, 508 Briggs, William L., 470 Brightman, Tom, 60 Brown, W. S., 498, 501 Brunet, Marie-Christine, 53, 488 Buchan, John, 62 q Buchanan, James L., 165 Buchholz, W., 60 Bukhberger, B., 305 Bulirsch, R., 83, 84, 190 Bunch, James R., 141, 146, 149, 219221, 225, 226, 231 q Buoni, John J., 198 Burgmeier, James W., 84 Businger, Peter A., 146, 193, 410 Butcher, J. C., 100 Byers, Ralph, 305, 320, 323, 324, 353, 567
N AME I NDEX Caffney, John, 520 Calvetti, D., 440442 Calvin (and Hobbes), 473 q Calve, M. P., 32 Campbell, S. L., 337 Campbell-Kelly, Martin, 245 q Canuto, Claudio, 32 Cao, Wei-Lu, 305 Caprani, Ole, 100 Cardano, Geronimo, 483 Carr III, John W., 57 Carter, Russell, 496 Cauchy, Augustin-Louis, 516 Cayley, Arthur, 446 Chaitin-Chatelin, Françoise, 53, 358, 488 Chan, Raymond H., 471 Chan, Tony F., 13, 33, 146, 147, 386 Chandrasekaran, Shivkumar, 136, 141, 386 Chang, Xiao-Wen, 225 Chartres, Bruce A., 190 Chatelin, Françoise, see Chaitin-Chatelin, Françoise Choi, Man-Duen, 513 q, 526 Cholesky, André-Louis, 224 Christiansen, Søren, 146 Chu, Eleanor, 193 Chu, King-wah Eric, 323 Cipra, Barry A., 32 Clasen, B.-I., 284 Clenshaw, C. W., 32, 53, 113 Cline, Alan K., 289q, 297, 299, 411, 480 Cline, R. E., 422 Clinger, William D., 61 Cody, Jr., William J., 39q, 54, 55, 59, 60, 495, 497, 499, 504 Cohen, A. M., 180, 522 Concus, P., 257 Corm, Andrew R., 299 Conte, Samuel D., 190 Cooley, James W., 470 Coomes, Brian A., 32 Coonen, Jerome T., 59, 495 Cope, J. E., 145 Coppersmith, Don, 448
667
N AME I NDEX
.
Corless, Robert M., 32 Cortés, Joaquín, 197 Cottle, Richard W., 224 Cox, M. G., 32 Crout, Prescott D., 195 Cryer, Colin W., 180, 181, 196 Curtis, A. R., 197 Cybenko, George, 225 Dahlquist, Germund, 83, 160 Daniel, J. W., 385 Datta, Karabi, 315 Davis, Philip J., 32, 100, 471 Dax, Achiya, 226, 337 Day, Jane M., 197 de Boor, Carl, 32, 190, 196 de Jong, Lieuwe Sytse, 33 de Rijk, P. P. M., 410, 411 Dekker, T. J., 57, 93, 284, 504 del Ferro, Scipione, 483 Demeure, Cédric J., 441 Demmel, James W., 47, 53, 60, 61, 84, 126, 140, 141, 143, 149, 165, 198, 207, 208, 224, 243, 248, 250-253, 257, 304,308, 322, 352, 417, 422, 486, 495, 499, 504, 527, 536 Dennis, Jr., J. E., 32, 328, 477, 478 Descloux, J., 343 Dhillon, Inderjit, 60, 536 Diamond, Harold G., 62 Dixon, John D., 300 Dongarra, Jack J., 188, 195,225,231 q, 257, 499, 581 q Doolittle, Myrick Hascall, 195 Dorn, William S., 83, 100, 113 Douglas, Craig C., 460, 575 Douglas, Jr., Jim, 32, 188 Doyle, Sir Arthur Conan, 289 q Drake, J. B., 261 q Zlatko, 225 Du Croz, Jeremy J., 265, 272, 273, 284 Dubrulle, Augustin A., 285, 511 Duff, Iain S., 143, 195, 200, 225, 226, 257, 304, 343, 410, 411, 527 Duncan, Martin, 96
Dunham, C. B., 425q Dwyer, Paul S., 169 q Eckart, Carl, 126 Edelman, Alan, 63, 162, 180, 181, 197, 198, 287, 386, 483n, 518, 518 q, 527, 534 Eirola, Time, 33 Eldén, Lars, 412 Eldersveld, Samuel K., 257 Elfving, Tommy, 441 Emel’yanenko, G. A., 305 Enright, Wayne H., 33 Erisman, A. M., 190, 193, 200 Espelid, Terje O., 100 Faddeeva, V. N., 195 Fairgrieve, Thomas F., 61, 530 Fan, Ky, 386 Farebrother, R. W., 409 Farkas, I., 113 Farnum, Charles, 497 Fateman, Richard J., 59 Feingold, David G., 257 Feldstein, A., 62 Ferguson, H. R. P., 461, 482 Ferguson, Jr., Warren E., 49, 60 Ferng, William R., 301 Fike, C. T., 114 Fischer, Patrick C., 460 Flannery, Brian P., 479, 490, 507 Fletcher, R., 32, 147, 149, 225, 226 Forsgren, Anders, 225 Forsythe, George E., 32, 33, 35, 52, 57, 84, 95, 138, 146, 164, 190, 196, 197, 235, 241, 242, 245 q, 261 q, 262, 282, 305, 325 q, 491 q, 526 Foster, Leslie V., 178, 411 Foulser, David E., 146, 147 Fox, L., xxviiq, xxviii, 35, 117 n, 188190 Frayssé, Valérie, 53, 358 Friedland, Shmuel, 348, 359 Funderlic, R. E., 198 Gaches, J., 132, 145
668 Gahinet, Pascal M., 323 Gal, Shmuel, 61 Gallopoulos, E., 488 Gander, Walter, 525 Gantmacher, F. R., 172 Gardiner, Judith D., 323 Gardner, Martin, 127 q Garner, Harvey L., 62 Gasca, M., 196 Gastinel, Noel, 123, 126 Gauss, Carl Friedrich, 1q, 195, 219, 325 q, 391 Gautschi, Walter, 425 q, 428, 429 Gautschi, Werner, 351 Gay, David M., 61 Geist, G. A., 261 q Geman, Stuart, 518 Gentle, James E., 34 Gentleman, W. Morven, 113,384,385, 470, 576 George, Alan, 193, 224, 226 Geuder, James C., 190 Ghavimi, Ali R., 320, 322 Gill, Philip E., 32, 225, 226, 229, 422, 479 Gill, S., 92 Givens, Wallace J., 33, 67 q Gluchowska, J., 411 Gohberg, I., 141, 440, 516 Goldberg, David, 39 q, 57, 93, 534 Goldberg, I. Bennett, 62 Goldstine, Herman H., 1 n, 33,52, 187, 196, 261q, 263, 517 Golub, Gene H., xxiv, 13, 27, 33, 146, 182, 190, 195, 223, 231 q, 257, 285, 301, 311, 312, 312 n, 327, 352, 384, 386, 388, 391, 392, 400, 409-412, 441, 580 Goodman, R., 62 Goodnight, James H., 285 Gould, Nicholas I. M., 181, 197 Govaerts, W., 242 Gragg, W. B., 297, 385 Graham, Ronald L., 87 q, 520 q Grebogi, Celso, 32 Greenbaum, A., 328, 329 Gregory, Robert T., 514, 525
N AME I NDEX Grimes, Roger G., 305, 527 Grosse, Eric, 581 q Gu, Ming, 352, 536 Gudmundsson, Thorkell, 301 Guggenheimer, Heinrich W., 287 Gulliksson, Mårten, 412 Gurwitz, Chaya, 32 Gustafson, John L., 459 Gustavson, F. G., 195 Hager, William W., 294, 304 Hall, Jr., Marshall, 179 Halmos, Paul R., 513 q Hamada, Hozumi, 54 Hammarling, Sven J., 322, 326, 327, 385, 503 Hammel, Stephen M., 32 Hammer, Rolf, 487 Hamming, R. W., 57, 443 Handy, Susan L., 412 Hansen, Per Christian, 146, 386, 461 Hanson, Richard J., 385, 409, 410, 422, 503 Harris, P. M., 32 Harter, Richard, 459 Hartfiel, D. J., 146 Hearon, John Z., 315 Heath, Michael T., 261 q, 399 Hein, Piet, 36 q, 127q Helvin, Marie, 166 q Henderson, Harold V., 323, 490 Hennessy, John L., 39 n, 57 Henrici, Peter, 32, 52, 53, 67q, 84, 351 Henson, Van Emden, 470 Heroux, Michael, 460, 575 Herzberger, Jurgen, 486 Hewer, Gary, 323 Higham, Desmond J., 32, 126, 141, 149, 178, 179, 195, 345q, 348, 407, 441 Higham, Nicholas J., 72, 100, 141, 145, 164, 165, 178, 179, 195, 197, 223, 225, 242, 243, 248, 250253, 257, 265, 272, 273, 284, 295, 300, 304-307, 322, 324, 328, 329, 354, 356, 358, 384,
N AME I NDEX 386, 389, 407, 410, 411, 417, 422, 440, 443, 460, 461, 489, 508, 574, 577 Hilbert, David, 526 Hildebrand, F. B., 33, 39 q Hocks, Matthias, 487 Hodel, A. Scottedward, 323 Hodges, Andrew, xxvii Hoffman, A. J., 386 Hoffmann, Christoph M., 34 Hoffmann, W., 284, 385 Hooper, Judith A., 504 Horn, Roger A., 119, 150, 310, 348, 546, 551, 555, 558, 580 Horning, Jim, 491 q Hotelling, Harold, 186, 187, 445 q Hough, David, 47, 509 Householder, Alston S., 2, 117q, 126, 160, 172, 383, 410, 565 Hull, T. E., 52, 61, 488, 506, 530 Huskey, H. D., 188 Hussaini, M. Yousuff, 32 Hyman, M. A., 282 Ikebe, Yasuhiko, 303, 305, 307 Incertis, F., 512 Ipsen, Ilse C. F., 136, 141, 147, 385, 386 Iri, Masao, 53, 488 Isaacson, Eugene, 189, 257 Jalby, William, 385 Jankowski, M., 94, 241, 242 Jansen, Paul, 486 Jennings, A., 165 Jennings, L. S., 421 Johnson, Charles R., 119, 150, 287, 310, 348, 546, 551, 555, 558, 580 Johnson, Samuel, 675 Jones, Mark T., 226 Jones, William B., 507 Jordan, Camille, 284 Jordan, T. L., 409 Jordan, Wilhelm, 284 Kågstrom, Bo, 305, 320, 322-324
669 Kahan, William M. (Velvel), 1q, 29, 33, 34, 46, 47, 47q, 50, 50q, 59, 63-65, 75, 86, 92-95, 98, 113, 123, 126, 136, 161, 165, 169 q, 225, 243, 486, 490, 494, 496, 497, 499, 501, 502, 507 Kahaner, David K., 391 q Kailath, T., 441 Kala, R., 323, 324 Kaniel, S., 226 Karasalo, Ilkka, 165 Karatsuba, A., 461 Karlin, Samuel, 523 Karlson, Rune, 404, 413 Karney, David L., 514, 525 Karp, A., 195 Karpinski, Richard, 56, 497 Kate, Tosio, 126 Kaufman, Linda, 221, 225, 226, 385 Keiper, Jerry B., 36 Keller, Herbert Bishop, 189,257,325 q Kennedy, Jr., William J., 34 Kenney, Charles S., 301, 323, 525 Kerr, Thomas H., 225 Andrzej, 84,94,224,242, 406, 411, 418, 422 Kincaid, D. R., 503 Kittaneh, Fuad, 525 Knight, Philip A., 328, 329, 354, 356, 358, 460, 461 Knuth, Donald E., xxiv, 54, 57, 58, 67q, 87q, 93, 94, 114, 461, 491 q, 520q, 526, 534 Koçak, Hüseyin, 32 Koltracht, I., 141, 517 Korner, T. W., 465q Kornerup, Peter, 54 Kostlan, Eric, 518, 527 Kovarik, Z. V., 145 Kowalewski, G., 440 Krasny, Robert, 505 Kreczmar, Antoni, 460 Krogh, F. T., 503 Krol, Ed, 582 Kruckeberg, F., 198 Kubota, Koichi, 488
670 J., 300 Kuki, H., 60 Kulisch, Ulrich W., 486, 487 Kuperman, I. B., 146 La Porte, M., 53 La Touche, Mrs., 87q, 88 Laderman, Julian, 450, 459 Lagrange, Joseph Louis, 219 Lancaster, Peter, 322 Lanczos, Cornelius, 490 Laratta, A., 421, 422 Larson, John L., 488 Larsson, S., 32 László, Lajos, 351 Laub, Alan J., 301, 320, 322, 323, 525 Läuchli, Peter, 388 Lawson, Charles L., 385, 409, 410, 422, 502 LeBlanc, E., 486 Lee, King, 460 Lehmer, D. H., 529 Lehoucq, R. B., 569 Lemeire, Frans, 165 Leuprecht, H., 98 LeVeque, Randall J., 13, 33 Lewis, John G., 33, 301, 305, 527 Li, T. Y., 285 Li, Xiaoye, 495 Linnainmaa, Seppo, 53, 83, 93, 100, 504 Linz, Peter, 100 Linzer, Elliot, 470 Liu, Joseph W. H., 224, 226 Longley, James W., 410 Lotstedt, Per, 412 Lotti, Grazia, 456, 457 Lu, Hao, 441 Lynch, Robert E., 32 Lyness, J. N., 32, 441 Lynn, M. Stuart, 327 Mac Lane, Saunders, 483 Macleod, Allan J., 197 Makhoul, John, 225
N AME I NDEX Malcolm, Michael A., 33, 95, 97, 196, 245q, 262, 305, 497, 509, 525 Marine, F., 461 Manteuffel, Thomas A., 165 Marovich, Scott B., 576 Marsaglia, George, 527 Martin, R, S., 196 Mascarenhas, Walter, 197 Mathias, Roy, 165, 223, 386, 515 Matsui, Shouichi, 53 Mattheij, R. M. M., 257 Matula, David W., 54, 57, 62 Mazzia, Francesca, 197 McCarthy, Charles, 146 McCracken, Daniel D., 83, 100, 113 McKeeman, William Marshall, 196, 241 McKenney, Alan, 322, 527 Meinguet, Jean, 224, 284 Mendelssohn, N. S., 286 Metcalf, Michael, 3 Metropolis, N., 489 Meurant, Gérard, 257, 305 Meyer, Jr., Carl D., 147, 337 Milenkovic, Victor J., 34 Miller, D. F., 322 Miller, Webb, 83, 115, 451, 454, 473q, 487, 488, 507, 517, 576 Miranker, Willard L., 486 Mirsky, L., 558 Moler, Cleve B., 33, 34, 59, 84, 95, 164, 190, 196, 197, 231 q, 235, 241, 243, 245q, 261q, 262, 282, 297, 304, 305, 323, 345q, 348, 359, 391q, 415q, 441, 503, 508, 511, 526, 579q Møller, Ole, 92 Montgomery, D., 186 Moore, Ramon E., 486 Morrison, Donald, 503, 511, 579 q Morton, K. W., 32 Mukherjea, Kalyan, 382 Müller, K. H., 113 Müller-Merbach, H., 32 Murakami, H., 483 n Murray, Walter, 32,225,226,422,479
N AME I NDEX Nagy, James G., 471 Nash, Stephen G., 312, 324, 391q, 567 Nashed, M. Zuhair, 409 Neumaier, A., 61, 94 Neumann, M., 198 Newbery, A. C. R., 113 Newcomb, Simon, 51 Newman, Morris, 514, 520 Nickel, Karl, 94, 486 Nilson, E. N., 167 Nordio, Marcelo, 197 Notay, Yvan, 329 Oberaigner, W., 98 O’Cinneide, Colm Art, 147 Oettli, W., 135, 145 Ofman, Yu., 461 O’Leary, Dianne Prost, 305 Olesky, D. D., 198 Oliver, J., 113, 440 Olkin, Ingram, 527 Olshevsky, V., 440, 441 Olver, F. W. J., 53, 76, 84, 113, 190 225 Opfer, Gerhard, 441 Ortega, James M., 384 Osborne, M. R., 421 Ostrowski, A. M., 350, 546 Paige, C. C., 225, 379, 385, 388, 397, 411, 415q, 421, 422, 570 Palmer, John, 525 Palmer, Kenneth J., 32 Pan, Victor, 441, 448, 450, 459 Papadimitriou, Pythagoras, 386 Papadopoulos, Philip M., 323 Park, Haesun, 385 Parlett, Beresford N., xxiv, xxviii, 27, 34, 58, 87q, 187, 219, 225, 361 q, 383-385 Pasternak, Mary E., 488 Patashnik, Oren, 87 q, 520 q Patrick, Merrell L., 226 Patterson, David A., 39 n, 57 Paxson, Vern, 61 Pelz, Richard, 505 Peña, J. M., 192, 196
671 Pereyra, Victor, 434, 436, 441 Peters, G., 113, 196, 279, 284, 287, 411 Peterson, Brian, 197 Philippe, Bernard, 385 Pichat, M., 98 Pierce, Daniel J., 301 Piessens, Robert, 32 Pinkus, Allan, 32, 190, 196 Plemmons, Robert J., 147, 198, 301, 337, 411, 422, 471, 579q, 580 Poljak, Svatopluk, 140 Polman, Ben, 257 Ponceleón, Dulce B., 226 Poromaa, Peter, 305, 320, 324 Pothen, Alex, 165 Powell, M. J. D., 225, 411, 474, 477 Power, Stephen, 224 Prager, W., 135, 145 Press, William H., 479, 490, 507 Priest, Douglas M., 33, 34, 58, 96-99, 102, 502, 504, 531 Pryce, J. D., 76, 242 Puglisi, Chiara, 371 Pukelsheim, Friedrich, 323 Puschmann, Heinrich, 197 Quarteroni, Alfio, 32 Quinlan, Gerald D., 96 Quinn, Thomas R., 96 Rabinowitz, Philip, 32, 100 Raimi, Ralph A., 52 Rail, Louis B., 409, 486, 488 Ramos, George U., 470 Rath, Wolfgang, 385 Ratz, Deitmar, 487 Ratz, H. C., 327 Razaz, M., 113 Reichel, Lothar, 113, 358, 440-442, 527 Reichelt, Mark W., 527 Reid, J. K., 3, 190, 193, 197, 200, 226, 411 Reinsch, C., xxviii, 284, 585 q Reiser, John F., 58
672 Ren, Huan, 60 Rew, R. K., 289 q, 299, 480 Rice, John R., 33, 385 Richtmyer, Robert D., 32 Rigal, J. L., 132, 145 Robertazzi, T. G., 98 Roberts, J. D., 322 Rohn, 128, 140, 141, 149 Romani, Francesco, 328 Rose, Donald J., 257 Rosenthal, Peter, 322 Ross, D. R., 97 Rowan, Thomas Harvey, 488, 489 Rubin, Donald B., 391 q, 410 Ruhe, Axel, 385 Ruiz, Daniel, 343 Rust, B. W., 145 Rutishauser, Heinz, 514, 520 Sameh, Ahmed H., 162, 164, 488 Samelson, Klaus, 57 Sande, G., 470 Sanz-Serna, J. M., 32, 33 Saunders, Michael A., 225, 226, 229, 257, 422 Sautter, Werner, 190 Scarborough, James B., 33, 58 Schaumburg, Kjeld, 305 Schelin, Charles W., 61 Scherer, R., 84 Schnabel, Robert B., 32 Schneider, Hans, 124, 359 Schonfelder, J. L., 113 Schreiber, Robert S., 165, 180, 245 q, 250–253, 257, 371, 384, 576 Schryer, N. L., 498 Schwartz, S. C., 98 Schwetlick, Hubert, 84,406,411,418, 422 Searle, Shayle R., 323, 490 Sha, Xuan-He, 450, 459 Shampine, Lawrence F., 32, 33, 84, 95, 527 Shannon, Claude E., 60 Shapiro, Alexander, 146 Shepherd, David, 59 Shinnerl, Joseph R., 225, 229
N AME I NDEX Shroff, Gautam M., 301 Shub, Michael, 147, 518 Simon, Horst D., 460 Skeel, Robert D., 36, 135, 146, 190, 192, 198, 235, 240, 241, 486 Slishman, Gordon, 460, 575 Smale, Steve, 2 n Smith, David M., 505 Smith, Francis J., 113, 439 Smith, Jon M., 37 Smith, Robert L., 503, 509 Smith, Roger M., 460, 575 Smoktunowicz, Alicja, 94, 141, 149, 242, 411 Snyder, James N., 231 q, 241 Sokolnicka, Jolanta, 242 Sorensen, Danny C., 195, 225, 229, 257 Sørevik, T., 461 Spellucci, P., 32 Spencer, Herbert, 665 Spieß, J., 460 Spooner, David, 488 Steele, Jr., Guy L., 61 Stegun, Irene A., 35 Sterbenz, Pat H., 33, 34, 50, 57, 60, 489 Stewart, G. W. (Pete), xxiv, 75, 113, 119, 126, 131q, 139, 146, 147, 151 q, 164, 174, 195, 197, 198, 224, 231 q, 241, 242, 297, 305, 312, 324, 359, 381,382,385,386,392, 394, 407,409-411,504, 519,527, 580 Stewart, William J., 305 Steer, J., 83, 84, 119, 126, 127, 190 Stone, Betty Jane, 126 Storey, C., 323 Stoutemyer, David R., 489 Strakos, Zdenek, 329 Strang, Gilbert, 15, 124, 146, 471 Strassen, Volker, 446, 461, 462 Straus, E. G., 138, 146 Street, Anne Penfold, 179 Stummel, Friedrich, 34, 83, 191
N AME I NDEX Sun, Ji-guang, 119, 126, 131 q, 139, 146, 194, 198,209,224,382, 386,392,394,404,407, 409, 411, 413, 580 Sun, Xiaobai, 384 Swarm, W. H., 474 Swartzlander, Jr., Earl E., 54 Sweeney, D. W., 60 Swenson, J. R., 52 Sylvester, James Joseph, 309 q, 322, 446 Tang, Ping Tak Peterj 61, 499, 530 Tang, W. P., 441 Tao, Pham Dinh, 128, 304 Tartaglia, Niccolo, 483 Taussky, Olga, 513 q, 515 Teukolsky, Saul A., 479, 490, 507 Thacher, Jr., Henry C., 196 Thisted, Ronald A., 34 Thompson, Sir D’arcy Wentworth, 1q Thron, W. J., 507 Tienari, Martti, 53 Todd, John, 126, 514, 520, 526, 527 Toh, Kim-Chuan, 483 n, 526 Torczon, Virginia J., 477-479 Tornheim, L., 180, 181 Traub, J. F., 440 Trefethen, Lloyd N., 6 q, 32, 169 q, 180, 328, 345 q, 346, 348, 352, 353, 355, 358, 483 n, 526, 527 Tremaine, Scott, 96 Tropp, Henry S., 491n Trummer, Manfred R., 32, 346, 355, 358 Tsao, Nai-kuan, 84, 384 Tukey, John W., 60, 470 Turing, Alan Mathison, xxvii, xxviii, 33, 126, 131 q, 188, 284,485 contributions in 1948 paper “Roundingoff errors...”, 188, 284 Turnbull, H. W., 383, 523 Turner, Kathryn, 225 Turner, Peter R., 51, 53, 165 Tyrtyshnikov, Evgenij E., 526
673 Underhill, L. G., 527 Ungar, Peter, 461 Urabe, Minoru, 62 Uspensky, J. V., 490 M., 461 van der Sluis, A., 137, 138, 198, 207, 391 q, 409 Van der Vorst, Henk A., 195, 225, 257, 329 Van Loan, Charles F., xxiv, 27, 141, 149, 182, 190, 195, 223, 225, 229, 231 q, 257, 299, 311, 312, 312n, 345q, 348, 352, 359, 370, 371, 384, 386, 388, 392, 409, 412, 465q, 466, 470, 580 van Veldhuizen, M., 196 van Wijngaarden, A., 501 Vanderbei, Robert J., 229 Varah, James M., 146, 167, 257, 258 Varga, Richard S., 160, 167, 257, 508 Vemulapati, Udaya B., 301, 412 225 Vetterling, William T., 479, 490, 507 Vieta, Franciscus, 483 Vignes, J., 53 Vitasek, Emil, 100 Viten’ko, I. V., 95 von Neumann, John, 33, 52, 186, 187, 196, 263, 517 Waite, William, 497, 499 Waldén, Bertil, 404, 413 Walker, Homer F., 32, 328 Wallis, Jennifer Seberry, 179 Wallis, W. D., 179 Wang, Johnson J. H., 198 Ward, Robert C., 285 Warming, Robert F., 527 Wasilkowski, G. W., 147 Wasniewski, Jerzy, 305 Wasow, Wolfgang R., 32 Watson, G. A., 146 Watterson, Bill, 473 Wedderburn, J. H. M., 117q
674 Wedin, Per-Åke, 392, 407, 409, 412, 422 Wegert, Elias, 353 Wei, Musheng, 388, 410 Weidnerj Peter, 486 Weiss, N., 147 Wendroff, Burton, 189, 197 Werner, Wilhelm, 113 Westin, Lars, 305, 323 Westlake, Joan R., 514 Wette, Matthew R., 323 White, Jon L., 61 Wichmann, B. A., 499, 502 Wilkinson, J. H., xxi, xxiv, xxvii, xxviii, 14, 25, 31, 33, 34, 39q, 52, 56, 58, 67q, 73, 75, 84, 100, l03q, 104, 113, 117q, 145, 151 q, 164, 175, 177, 178, 180-182, 187-191, 195-197, 203 q, 207, 208, 224, 232, 241, 252, 279, 283-285, 287, 297, 326, 327, 343 q, 348, 361 q, 364, 365, 384, 385, 387, 391, 400, 409-411, 474, 485, 542, 585 q first program for Gaussian elimination, 195–196 on the purpose of a priori error analysis, 203 q solving linear systems on desk calculator in 1940s, 187–188 user participation in a computation, 31 Williams, Jack, 425 q Wilson, Greg, 60 Winograd, Shmuel, 446, 448, 461 Wisniewski, John A., 488 Witzgall, C., 119, 126, 127 Wolfe, Jack M., 97 Wolfe, Philip, 32 Woodger, Michael, 169 q, 445 q H., 32, 94, 147, 241, 242, 300, 328 Wrathall, Celia, 488 Wright, Margaret H., 32, 225, 479 Wright, Stephen J., 178
N AME I NDEX Yalamov, P. Y., 83 Yip, E. L., 490 Yohe, J. Michael, 486, 504 Yorke, James A., 32 Young, David M., 326 t, 350 Young, Gale, 126 Yuvzd, Gideon, 459 Zang, Thomas A., 32 Zawilski, Adam T., 328 Zehfuss, Johann Georg, 323 Zeller, K., 84 Zeng, Z., 285 Zha, Hongyuan, 167, 382 Zielke, Gerhard, 126, 527 K., 323 Ziv, Abraham, 60, 61, 83, 84 Zlatev, Zahari, 305
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Subject Index Knowledge is of two kinds. We know a subject ourselves, or we know where we can find information upon it. — SAMUEL JOHNSON, Boswell’s Life of Johnson (1775)
A suffix “t” after a page number denotes a table, “f” a figure, “n” a footnote, and “q” a quotation. Mathematical symbols and Greek letters are indexed as if they were spelled out. The solution to a problem is not indexed if the problem itself is indexed under the same term.
Aasen’s method, 226, 229 absolute error, 4 absolute norm, 119 accuracy versus precision, 7, 33 ACRITH, 486 Aitken extrapolation, 101 alternating directions method, 477 approximation theory, references for rounding error analysis, 32 Augment precompiled, 486, 504, 506 augmented system matrix, 393 scaling and conditioning, 402 Automatic Computing Engine (ACE), 56, 189t, 195, 343 automatic differentiation, 488 automatic error analysis, 473–490, see also interval analysis; running error analysis condition estimation, 480-481 solving a cubic, 483–484 Strassen’s inversion method, 481482 using direct search optimization, 474-477
componentwise, 134, 141 evaluating, 143 componentwise relative, 134 definition, 7 least squares problem, 404-407, 413 linear system Oettli-Prager theorem, 135 Rigal-Gaches theorem, 14, 132 Lyapunov equation, 316–317 mixed forward-backward error, 8 normwise, 132 normwise relative, 132 preserving symmetric structure, 149, 406–407 structured, 141 Sylvester equation, 313-316 underdetermined system, 419, 423 backward error analysis development by Wilkinson, 33– 34, 189 in differential equations, 33 motivation, 7 not a panacea, 1q purpose, 71–72, 203 q
backward error, 7-8
675
Home
676 backward stability componentwise, 142 definition, 8 normwise, 142 banded matrix, growth factor, 183 Bartels-Stewart method, 311-313 Bauer’s scaling theorem, 139, 146 bilinear noncommutative matrix multiplication algorithm, 449– 450 error analysis, 456–457 binary-decimal conversion, 61-62 bit, 60 BLAS (Basic Linear Algebra Subprograms), 586-587 fast level 3, 460 level 2 extended precision extension, 506 xNRM2 (2-norm), 502–503, 510– 511 block algorithm advantages of, 245 q definition, 246 block diagonal dominance, 251-255, 257 and block LU factorization, 252– 255 definition, 251 block LU factorization, 246-259 computation, 247 definition, 246 error analysis, 250-257 existence and uniqueness, 247 stability for (point) diagonally dominant matrix, 255 for block diagonally dominant matrix, 25 1–255 for block tridiagonal matrix, 258 for symmetric positive definite matrix, 255–257 Bunch-Kaufman factorization, 221-223 Bunch-Parlett factorization, 219-220 byte, 60 calculator, displaying words on, 37
S UBJECT I NDEX cancellation, 10–1 1, 30 in summation, 91, 543 not a bad thing, 11 of rounding errors, 21–26 Cauchy matrix, 516-517 inverse, 516 LDU factors, 516-517 Cauchy-Schwarz inequality, 119 CELEFUNT, 499 chaos, references for rounding error, analysis, 32 Chebyshev spectral differentiation matrix, 346, 355 Cholesky factorization, 204 computation of, 205 conditions for success in floating point, 208-209 error analysis, 205–209 existence and uniqueness, 204 perturbation bounds, 209–210 semidefinite matrix complete pivoting, 211 computation of, 210–21 1 error analysis, 214–218 existence and uniqueness, 210 perturbation theory, 21 1–214 termination criteria, 21 7–218 chopping, 57 circulant matrix, 468 circulant system, error analysis for solution by FFT, 468-470 CLAPACK, 588 colon notation, 2–3 companion matrix, 525-526 singular values, 525–526 comparison matrix, 157 compensated summation, 92–97 complete pivoting, 170 early use of, 196 fallacious criticism of, 200 growth factor, 180-181, 197 conjecture proved false, 181 complex arithmetic, error analysis, 78– 80, 84 complex number division without overflow, 503504, 509
S UBJECT I NDEX square root of, 36 componentwise relative error, 5 condition number distance to singularity and, 123, 126, 140 estimation, 289–308 counterexamples, 289 q, 294– 296, 299, 304, 305 counterexamples by direct search, 480-481 for tridiagonal matrices, 301– 303 incremental, 301 LAPACK estimator, 294-297, 480481 LINPACK estimator, 297-299 probabilistic methods, 300-301 general theory, 33 Hadamard, 281, 287 minimizing by scaling, 136–139, 146 of function, 9 of linear system componentwise, 135 normwise, 133 of rectangular matrix, 392 of square matrix, 121–123, 126 of summation, 100 Skeel’s, 135 conjugate gradient method, 328, 329, 341 circulant preconditioned, 471 continued fraction algorithms and error analysis, 507 evaluating in IEEE arithmetic, 492-493 running error bound, 85 convergent matrix, 348 conversion, binary-decimal, 61–62 correct significant digits, 4–5, 32 Cramer’s rule, (instability of, 14-15, 34, 37 Cray computers adoption of IEEE arithmetic, 49 arithmetic on, 39 q, 495 puzzling results from Cray Y-MP and Cray 2, 496497
677 UNICOS library, 448, 450 Crout’s method, 174 CS decomposition, 388, 408 cubic equation Newton’s method, 489-490 stability of explicit formulae for roots, 483484 cyclic reduction, 197 denormalized numbers, see subnormal numbers departure from normality (Henrici’s), 351-352 determinant, 281-283 computation of, 282–283 condition number of, 287 of upper Hessenberg matrix, 27– 28 diagonal dominance, 181 and block LU factorization, 255 block, 251-255, 257 bound for LU factors of tridiagonal matrix, 185 growth factor, 181 matrix inverse bound, 167 for tridiagonal matrix, 303 diagonal pivoting method, 218-223 complete pivoting and its stability, 219–220 growth factor complete pivoting, 220 partial pivoting, 222 partial pivoting and its stability, 221-223 diagonally dominant matrix bound for inverse, 167 growth factor for, 181 differential equations, see ordinary differential equations; partial differential equations, references for rounding error analysis direct search optimization methods, 477-479 discretization error, 6 distance to singularity componentwise, 140
678 normwise, 123 divided differences, 109-112 confluent, 430 Doolittle’s method, 173-174, 195 double rounding, 48, 63, 541 Drazin inverse, 336-337 drift, in floating point arithmetic, 58 dual norm, 119 dual vector, 119 dynamical systems, references for rounding error analysis, 32 effective conditioning, 146 EISPACK, 587 ELEFUNT package, 499 equilibration, 136, 138, 192 error absolute, 4 backward, see backward error forward, see forward error mixed forward-backward error, 8 relative, 4, 5 sources of, 5-6 error analysis, see rounding error analysis ESSL library (IBM), 448, 450 expm 1 function (ex – 1), 34 fan-in algorithm, 165 for summation, 88 for triangular system solution, 162– 164 fast Fourier transform, 465-471 Cooley-Tukey factorization of DFT matrix, 466 error bound, 467 for solving circulant systems, 468470 fast matrix multiplication, 445463 bilinear noncommutative algorithm, 449450 deriving methods, 459460 error analysis, 450-459 in the level 3 BLAS, 460 Miller’s error results, 451 record exponent, 448 Strassen’s method, 446448
S UBJECT I NDEX Winograd’s variant, 448, 455– 456 3M method for complex multiplication, 450 Winograd’s method, 446 fixed point arithmetic, 56 fl operator (rounding), 42 floating point arithmetic, 39-65 alternatives to, 53-54 banned from safety-critical systems, 499 binary-decimal conversion, 61-62 choice of base, 51, 60–61 compiler optimization, dangers of, 497 determining properties of, 497498 drift in, 58 earliest subroutines, 39 q formal algebra, 58 fused multiply-add operation, 60, 65 IEEE arithmetic, see IEEE arithmetic Language Independent Arithmetic (LIA-1), 502 model, 58, 501–502 Brown’s, 498, 501-502 standard, 44 with underflow, 61 without guard digit, 49 multiple precision, 504–506 parameters for selected machines, 41t parameters in software, specifying, 499–500 representation error, 51 rounding, see rounding software issues, 491–512 speed of operations (relative), 60 subnormal numbers, 41, 47, 495 subtraction done exactly, 49–50 testing accuracy of, 54–56 testing correctness of, 498499 unit roundoff, 3, 42 wobbling precision, 43, 51 floating point coprocessor, 47
S UBJECT I NDEX floating point numbers characterization, 40 normalized, 40 spacing between, 41 subnormal, 41, 47 testing for equality, 495 flop, 3 Fortran 90, 3 environmental inquiry functions, 498 matmul, 460 forward error, 7–8 definition, 7 for linear system, 13 mixed forward-backward error, 8 forward stability componentwise, 142 definition, 10 normwise, 142 Fourier matrix, 179 FPV (floating point verification) package, 498499 Frobenius norm, 120 fused multiply-add operation, 60, 65 γ n (error constant) definition, 69 properties, 74 Gauss-Jordan elimination, 275-281,284285 algorithm, 276 error analysis, 277–281 Gauss-Seidel method, 325 q, 334 Gaussian elimination, 170–174, see also LU factorization a posteriori stability tests, 192– 194 complete pivoting, 170 computer programs first, 195-196 history of, 196 connection with LU factorization, 171 error analysis, 174–177 history of, 186-191 growth factor, 177–183, see also growth factor
679 in ancient China, 195 loop orderings, 195 need for pivoting, 170 on Hessenberg matrix, 27–28 partial pivoting, 170, 173 pessimism of its accuracy in 1940s, 186–187 row and column scaling, 191–192 threshold pivoting, 200 use by Gauss, 195 versus Cramer’s rule, 14–15 without pivoting, instability of, 17 Gelfand’s problem, 51 geometric computation, accuracy of algorithms in, 34 Givens rotation, 371 disjoint rotations, 373-375, 387 fast, 385 gradual underflow, 47, 61 Gram-Schmidt method, 376-381 classical algorithm, 377 error analysis, 378–379, 381 modified algorithm, 377 connection with Householder QR factorization, 361q, 379, 385 error analysis, 378–381 error analysis for application to LS problem, 396-397 stability, 27 reorthogonalization, 385 growth factor, 177-183 a posteriori estimates for, 193 define using exact or computed quantities?, 177, 196 for banded matrix, 183 for complete pivoting, 180-181 for diagonal pivoting method, 220, 222 for diagonally dominant matrix, 181 for partial pivoting, 177-183 large growth in practical problems, 178
680 for random matrices, 196-197 for tridiagonal matrix, 183 for upper Hessenberg matrix, 182 lower bound for, 179 maximization by direct search, 475– 476 numerical maximization for complete pivoting, 181, 197 statistical model of, 180 guard digit, 48 test for, 56 Haar distribution, random orthogonal matrix from, 519–520 Hadamard condition number, 281,287 Hadamard matrix, 128, 179, 181, 201 Hadamard’s inequality, 287 Harwell-Boeing sparse matrix collection, 527 Heron’s formula, 50 Hessenberg matrix determinant of, 27-28, 34, 282283 Gaussian elimination, 27-28, 34 growth factor for, 182 Hewlett-Packard HP 48G calculator condition estimator, 304 exhausting its range and precision, 17–18 hidden bit, 45 Hilbert matrix, 514-517 Cholesky factor, 515 inverse, 515 Holder inequality, 118, 119 Homer’s method, 104-115 for derivatives, 106-109 for rational function, 29 running error bound, 105–106, 113 Hough’s underflow story, 509-510 Householder transformation, 362-363 aggregated (WY representation), 370-371 block, 384 error analysis, 364–369 history of, 383 in QR factorization, 363–364 Hyman’s method, 34, 282-283, 287
S UBJECT I NDEX IBM, ESSL library, 448, 450 IEEE arithmetic, 43, 4548 double rounding, 48, 63, 541 exception handling, 46, 493–495 exceptions, 4647 exploiting in software, 492–495 extended formats, 47 gradual underflow, 47 implementation using formal methods, 59-60 46–47, 492, 495 NaN, 46, 492, 495 parameters, 41 t, 45 recommended auxiliary functions, 495 rounding modes, 46 signed zeros, 46 Standard 754, 45 Standard 854, 48 subnormal numbers, 47, 495 index of a matrix, 336 (IEEE arithmetic), 46–47, 492, 495 inner product error analysis, 68–71 in extended precision, 70 reducing constant in error bound, 69 Intel Pentium chip, division bug, 59 Internet, 582 interval analysis, 46, 198, 485–487 dependencies, 485 fallibility, 487 Gaussian elimination, 485-486 super-accurate inner product, 486 interval arithmetic, see interval analysis inverse iteration, 27 inverse matrix, 26 1–287 bound using diagonal dominance, 167 error analysis for Gauss-Jordan elimination, 277–281 for LU factorization, 270-275 for triangular matrix, 265-270 high accuracy computation of, 284 in solving Ax = b, stability, 262
S UBJECT I NDEX left and right residuals, 263-264 perturbation theory, 140 times for computation on early computers, 276 t triangular bounds for, 159-161 error analysis, 265–270 why not to compute, 262 involutary matrix, 521 iterative methods, see also stationary iterative methods dates of publication, 326 t error analysis, 329-341 survey of, 327–329 stopping criteria, 341–342 iterative refinement, 30, 188, 231–244, 497 backward error analysis, 235-240 behaviour with GEPP, 239 with QR factorization, 375–376 condition number estimate from, 243 for least squares problem, 399403, 410 for Vandermonde system, 438– 440 forward error analysis, 232-235 in fixed precision, 234 in mixed precision, 234 LAPACK convergence test, 242 practical issues, 242–243 Jacobi method, forward error analysis, 332–334 Jordan canonical form, 346-348 Kahan matrix, 161, 214 second smallest singular value, 167 Kreiss matrix theorem, 353 Kronecker product, 310, 323 LANCELOT, 181, 197 LAPACK, 587-589 block and partitioned LU factorization, 258 Cholesky factorization, 226-227
681 condition number estimation, 294– 297, 306 diagonal pivoting method, 227 forward error bound for linear systems, 144 iterative refinement, 242–243 least squares problem, 412 LU factorization, 192, 198-199, 258 matrix l-norm estimator, 294– 297, 480–481 matrix inversion, 285 QR factorization, 386-387 Sylvester equation, 324 test matrix generation, 527 triangular systems, 166 2 x 2 problems, solving, 500-501 underdetermined system, 423 xLAMCH for determining machine parameters, 498 Latin, neoclassic, publishing papers in, 470 LDLT factorization, 205 block, 218 least significant digit, 40 least squares problem, 391-413 augmented system matrix, 393 scaling and conditioning of, 402 backward error, 404-407, 413 constrained, 412 iterative refinement, 399–403, 410 Longley test problem, 410 modified Gram–Schmidt, error analysis, 396–397 normal equations error analysis, 397–399 versus QR factorization, 399 perturbation theory, 392–394 QR factorization, error analysis, 395-396 seminormal equations, 403–404 weighted, 411–412 Leja ordering, 111, 113, 115, 438 level index arithmetic, 53 linear system large dense, in applications, 198 perturbation theory, 131–150
682 practical forward error bounds, 143–144 records for largest solved, 199 t scaling before Gaussian elimination, 191–192, 197–198 times for solution on early computers, 189 t LINPACK, 587 Cholesky factorization of semidefinite matrix, 217 condition estimator, 297–299, 305 iterative refinement, 242 LU factorization, 192 matrix inversion, 267, 271–274 tridiagonal system solution, 306 logarithmic distribution of numbers, 51, 53 Longley test problem, 410 LU factorization, 169-201, see also Gaussian elimination a posteriori stability tests, 192– 194 block, see block LU factorization complete pivoting, 170 Crout’s method, 174 determinantal formulae for factors, 172 Doolittle’s method, 173-174 error analysis, 174–177 history of, 186-191 existence and uniqueness, 171172 for nonsymmetric positive definite matrix, 223–224 growth factor, 177-183, see also growth factor loop orderings, 195 of Hessenberg matrix, 27–28 of tridiagonal matrix, 184 partial pivoting, 170, 173 partitioned, error analysis of, 248– 250 perturbation bounds, 194 row and column scaling, 191–192 stability for M-matrix, 198 versus Cramer’s rule, 14–15
S UBJECT I NDEX without pivoting, instability of, 17 Lyapunov equation backward error, 316-317 discrete-time, 322 M-matrix, 580 stability of LU factorization, 198 triangular, 157, 159, 160 machar code, 497498 machine epsilon, 41 magic square matrix, p norm of, 127 mantissa, 40 Maple, 6, 181, 506 Markov chain, perturbation analysis for, 146 Mathematical, 6, 181, 506 MATLAB, 3, 43, 583 condest, 304, 480 fft, 468 inv, 271 rand, 518 randn, 518 rcond, 305, 480, 489 roots, 483 special matrices, 514 Symbolic Math Toolbox, 3, 521 Test Matrix Toolbox, 591–594 matrix block diagonally dominant, 251 Cauchy, 516-517 circulant, 468 companion, 525–526 comparison, 157 condition number, 121–123, 126, 392 confluent Vandermonde-like, 429 convergent, 348 diagonally dominant, 181 distance to singularity, 123, 140 Drazin inverse, 336-337 Fourier, 179 Hadamard, 128, 179, 181, 201 Hilbert, 514–517 inversion, 261–287, see also inverse matrix involuntary, 521
S UBJECT I NDEX Kahan, 161, 167, 214 M-matrix, 580 magic square, 127 moment, 520 nonsymmetric positive definite, 223 Pascal, 520-524 powers of, 345–359, see also powers of a matrix pseudo-inverse, 392, 409, 412, 416 random, 517–520 randsvd, 519–520 second difference, 525 semiconvergent, 337 Sylvester’s introduction of term, 309 symmetric indefinite, 218 symmetric positive definite, 204 symmetric positive semidefinite, 210 symmetric quasidefinite, 229 test, 513–528 totally nonnegative, 176 tridiagonal, 183–186 Toeplitz, 524–525 Vandermonde, 426 Vandermonde-like, 429 vet-permutation, 319, 323 matrix multiplication backward error, 85 error analysis, 76–78, 85 fast methods, 445-463 matrix norm, see norm meaningless answers, why you might get them, 35 misconceptions, of floating point arithmetic, 31 mixed forward–backward error, 8 modified Gram–Schmidt method, see Gram–Schmidt method, modified moment matrix, 520 monotone norm, 119 Moore-Penrose conditions, 412 most significant digit, 40 multi-directional search method, 477– 479 multiple precision arithmetic, 504–506
683 Bailey’s package MPFUN, 505 Brent’s package, 486, 504 mutation testing, 517 NAG Library, 583-584 LAPACK in, 588 machine constants, 500 NaN (not a number), 46, 492, 495 Nelder-Mead simplex method, 479 netlib, 582–583 nonlinear equations, references for rounding error analysis, 32 nonsymmetric positive definite matrix, 223 LU factorization, stability of, 223224 norm, 117–129 || · ||α,B, explicit formulae for, 128 absolute, 119 consistent, 120 dual, 119 Frobenius, 120 Holder inequality, 118 matrix, 120–124 norm equivalence constants, 122t matrix p-norm, 124–126 of magic square matrix, 127 monotone, 119 subordinate matrix, 120, 121 2-norm, evaluation without overflow, 502-503 unitarily invariant, 121 vector, 118–119 norm equivalence constants, 121t normal equations, error analysis, 397– 399 notation, explanation of, 2–3, 73–76 0°, definition, 64 NPSOL, 197 numerical analysis, definition, 6 numerical radius, 350 numerical stability definition, 8, 33 for linear equation solvers, 141142
684 Oettli–Prager backward error theorem, 135 optimization, references for rounding error analysis, 32 ordinary differential equations accuracy of mesh point formation, 101 backward error in, 33 Euler’s method with compensated summation, 95 references for rounding error analysis, 32 outer product, error analysis, 71 overflow, 18, 42 avoiding, 502–504 p-norm power method, 291–294, 304 parallel prefix operation, 165 paranoia code, 497498 partial differential equations, references for rounding error analysis, 32 partial pivoting, 170, 173 early use of, 196 growth factor, 177-183 large growth in practical problems, 178 threshold pivoting, 200 partitioned algorithm, definition, 246 partitioned LU factorization, error analysis, 248–250 Pascal matrix, 520-524 Cholesky factor, 521 inverse, 522 total positivity, 523 Patriot missile software problem, 506– 507 Pentium chip, division bug, 59 performance profile, for LAPACK norm estimator, 296–297 perturbation theory by calculus, 144-145 linear systems, 131-150 statistical, 147, 149-150 Sylvester equation, 318-320 p i (π ), high precision calculation as computer test, 491q
S UBJECT I NDEX polar decomposition, 386, 389 polynomials, 103–1 15, see also Horner’s method divided differences, 109–1 12 fast evaluation schemes, 114, 115 Newton form, 109-112 PORT library, machine constants, 500 portability of software, 499-502 positive definite matrix, see nonsymmetric positive definite matrix; symmetric positive definite matrix power method, 26 for matrix l-norm estimation, 294297 for matrix p-norm estimation, 291294, 304 powers of a matrix, 345–359 behaviour of stationary iteration, 358 departure from normality, 351352 hump, 348-349 in exact arithmetic, 346–353 in finite precision arithmetic, 353– 358 pseudospectrum, 352, 356-358 role of spectral radius, 346–348 precision effect of increasing, 19-21 versus accuracy, 7, 33 program verification, applied to error analysis, 488 pseudo-inverse, 392, 409, 412, 416 ε-pseudospectral radius, 352 pseudospectrum, 352, 356-358 of companion matrix, 526 Pythagorean sum, 503, 511-512 QR factorization, 361-389 column pivoting, 387 Givens, 371-373 cancellation of errors in, 24–26 error analysis, 373–375 Householder, 363-364 error analysis, 368–369
S UBJECT I NDEX error analysis for application to LS problem, 395-396 error analysis for partitioned (WY representation), 370371 iterative refinement for linear system, 375–376 perturbation theory, 381–383 rank-revealing, 386 quadratic equation, solving, 1 1–12, 33 quadrature accuracy of grid formation, 101 error bound for evaluation of rule, 86 references for rounding error analysis, 32 random matrices, 5 17–520 condition number of, 518 expected number of real eigenvalues, 518 orthogonal, 519, 527 spectral radius of, 518 tend to be well conditioned, 518 2-norm of, 518 with given singular values, 519– 520 randsvd matrix, 519–520, 527 range reduction, 54 RCOND condition estimator (LINPACK, MATLAB), 305, 480–481, 489 relative error, 4, 5 componentwise, 5 relative error counter, , 75 relative precision, 76 relative residual, 14 research problems, 102, 115, 201, 229, 244, 287, 308,324,359,423, 442, 463, 490, 527, 528 residual, relative, 14 Riccati equation, algebraic, 322 Rigal–Gaches backward error theorem, 132 rounding, 4, 42 dealing with ties, 42, 58 to even versus to odd, 58 rounding error analysis
685
automatic, 473–490 demystified, 82–83 graphs in, 83 model standard, 44 with underflow, 61 without guard digit, 49 notation, 73–76 ordering of operations, effect of, 77, 154 purpose of, 71–72, 203 q statistical approach, 52–53 rounding errors accumulation of, 16 are not random, 29, 52 beneficial effects of, 26-27 cancellation of, 21–26 in subtraction, 49–50 statistical assumptions on, 52–53 rules of thumb condition for computed powers of matrix to converge to zero, 358 forward error related to backward error and condition number, 10 relative speed of floating point operations, 60 square root of constants in error bound, 52 Runge-Kutta method, 92, 100 running error analysis, 73, 489 for continued fraction, 85 for Homer’s method, 105–106, 113 for inner product, 72–73 sample variance, see variance ScaLAPACK, 588 scaling a linear system before Gaussian elimination, 191-192, 197– 198 scaling to minimize the condition number, 136-139, 191 Schur complement, 219, 224, 247, 252 perturbation bounds for symmetric positive semidefinite matrix, 212–218
686 second difference matrix, 525 semiconvergent matrix, 337 seminormal equations for least squares problem, 403404 for underdetermined system, 417 separation (sep), of two matrices, 318 Sherman-Morrison formula, 197, 490 significance arithmetic, 489 significant digits correct, 4–5, 32 least and most significant, 40 singular value decomposition (SVD), 580 software avoiding underflow and overflow, 502-504 effects of underflow, 504 issues in floating point arithmetic, 491-512 portability, 499-502 specifying arithmetic parameters, 499-500 specifying numerical constants, 501 SOR method, forward error analysis, 334 square root, of complex number, 36 stable algorithms, designing, 30-31 stationary iterative methods, 325–343 and powers of a matrix, 358 backward error analysis, 334-336 forward error analysis, 329-334 singular systems, 338–341 Jacobi method, 332-334 scale independence, 331 singular systems, theory for, 336– 338 SOR, 334 stopping criteria, 341–342 statistics, see also variance computational references, 34 sticky bit, 46 Strassen’s method, 446-448 accuracy compared with conventional multiplication, 454455 error analysis, 452–456
S UBJECT I NDEX error versus operation count, 16 for inversion, 461-462, 481-482 implementation issues, 460 Winograd’s variant, 448,455-456 subdifferential, of a vector norm, 291 subgradient, 292 subnormal numbers, 41, 47, 495 summation, 87–102 choice of method: summary, 98– 100 compensated and applications, 92– 97 condition number, 100 criterion for minimizing error, 90 distillation algorithms, 98 doubly compensated, 96-97 error analysis, 89–92 insertion method, 88 pairwise (fan-in), 88 recursive, 88 ordering in, 19, 90-91 statistical error estimates, 98 SVD (singular value decomposition), 580 Sylvester equation, 309-324 backward error, 313-316 Bartels-Stewart method, 311-313 generalizations, 321–324 perturbation theory, 318-320 practical error bounds, 320–321 solution methods, 311–313 symbolic manipulation package, 6 symmetric indefinite factorization, 218, see also diagonal pivoting method symmetric indefinite matrix, 218 symmetric positive definite matrix, 204 and block LU factorization, 255– 257 practical test for, 225 symmetric positive semidefinite matrix, 210 determinantal conditions for, 228 symmetric quasidefinite matrix, 229 synthetic division, 107 tablemaker’s dilemma, 5 test
S UBJECT I NDEX for accuracy of floating point arithmetic, 54–56 for guard digit, 56 test matrices, 513–528 Harwell–Boeing collection, 527 Test Matrix Toolbox, 591-594 3M method, 450, 461, 505 error analysis, 458–459 Toeplitz matrix pseudospectra, 527, 528 f tridiagonal, 524–525 totally nonnegative matrix, 176, 523 LU factorization, 176, 196 row scaling in, 192 test for, 196 transformations, well conditioned, 30 transputer (Inmos), proof of arithmetic’s correctness, 59 triangular matrix bounds for inverse, 159–161 condition numbers, 155 inversion, 265–270 inversion methods blocked, 267–270 unblocked, 265–267 M-matrix, 157, 159, 160 triangular systems, 151–168 accurate solution of, 151 q, 155, 156, 159 conditioning, 156–157 fan-in algorithm, 162-164 partitioned inverse method, 165 substitution backward error analysis, 152– 154 forward error analysis, 155-159 tridiagonal matrix, 183–186 condition number estimation, 301– 303 growth factor, 183 LU factorization, 184 error analysis of, 184–186 structure of inverse, 303, 305 Toeplitz, 524-525 truncation error, 6 Turing Award of the ACM, xxvii, 59 Turing programming language, 506
687 Numerical Turing, 506 2 x 2 problems, reliable solution of, 500-501 ulp (unit in last place), 43 uncertainty, in data, 5 underdetermined system, 415–423 backward error, 419, 423 backward stability, definition, 419 modified Gram–Schmidt, 421–422 perturbation theory, 417–419 Q method (QR factorization), 416 error analysis, 419-422 seminormal equations, 417 error analysis, 421 underflow, 18, 42 avoiding, 502–504 effects on software, 504 model for error analysis, 61 UNICOS library (Cray), 448, 450 unit roundoff, 3, 42 update formula, involving small correction, 30 van der Sluis’s theorem, 137 Vancouver Stock Exchange, inaccurate index, 57–58 Vandermonde matrix bounds and estimates for condition number, 428 definition, 426 inverse, 426–428 inversion algorithm, 427 LU factorization in factored form, 432–433 QR factorization, 441 structured condition number, 440441 Vandermonde system, 425-443 accuracy independent of condition number, 436 algorithm for dual, 431–432 for primal, 433-434 for residual of confluent system, 439 backward error analysis, 437
688 complexity results, 441 curing instability, 438440 forward error analysis, 435-436 history of solution methods, 441 preventing instability, 438 Vandermonde-like matrix confluent, definition, 429 definition, 429 determinant, 442 variance algorithms for computing, 12–13, 33 condition numbers for, 37 error bound for two-pass formula, 38 vec operator, 310 vet-permutation matrix, 319, 323 Venus probe, loss due to program bug, 491 q Wedin’s least squares perturbation theorem, 392–393 proof, 407–409 Winograd’s method, 446, 461 error analysis, 451452 scaling for stability, 452 wobbling precision, 43, 51 WY representation of product of Householder matrices, 370-371 0°, definition, 64
S UBJECT I NDEX
What is the most accurate way to sum floating point numbers? What are the advantages of IEEE arithmetic? How accurate is Gaussian elimination, and what were the key breakthroughs in the development of error analysis for the method? The answers to these and many related questions are included in Accuracy and Stability of Numerical Algorithms. This book gives a thorough treatment of the behavior of numerical algorithms in finite precision arithmetic. It combines algorithmic derivations, perturbation theory, and rounding error analysis. Software practicalities are emphasized throughout, with particular reference to LAPACK and MATLAB. The best available error bounds, some of them new, are presented in a unified format with a minimum of jargon, and pertubation theory is treated in detail. Historical perspective and insight are given, with particular reference to the fundamental work of Wilkinson and Turing. The many quotations provide further information in an accessible format. The book is unique in that algorithmic developments and motivations are given succinctly and implementation details are minimized so that readers can concentrate on accuracy and stability results. Not since Wilkinson’s Rounding Errors in Algebraic Processes (1963) and The Algebraic Eigenvalue Problem (1965) has any volume treated this subject in such depth. A number of topics are treated that are not usually covered in numerical analysis textbooks, including floating point summation, block LU factorization, condition number estimation, the Sylvester equation, powers of matrices, finite precision behavior of stationary iterative methods, Vandermonde systems, and fast matrix multiplication. Nicholas J. Higham is Professor of Applied Mathematics at the University of Manchester, England. He is the author of more than 40 publications and is a member of the editorial boards of the SIAM Journal on Matrix Analysis and Applications and the IMA Journal of Numerical Analysis . His book handbook of Writing for the Mathematical Sciences was published by SIAM in 1993 ____________________________________________________________ For more information about SIAM books, journals, conferences memberships, or activities, contact:
SIAM Society for Industrial and Applied Mathematics 3600 University City Science Center Philadelphia, PA 19104-2688 USA Telephone: 215-382-9800 / Fax: 215-386-7999
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ISBN 0-89871-355-2
